THE 



ELASTICITY AND RESISTANCE 



OF THE 



MATERIALS OF ENGINEERING. 



BY. 



WM. H. BURR, C.E., 

WILLIAM HOWARD HART PROFESSOR OF RATIONAL AND TECHNICAL 
MECHANICS AT RENSSELAER POLYTECHNIC INSTITUTE. 






IC^« 







NEW YORK : 
JOHN WILEY & SONS. 

1883. 



^ 



Copyright, 1883, 
By WM. H. burr. 



PRESS OF i. i. LITTLE t CO., 
KOS. 10 TO 20 ASTOR PLACE, NEW YORK. 



PREFACE. 



This work has been the outgrowth of lectures on the 
elasticity and resistance of materials, given by the author to 
succeeding classes of students in the department of civil engi- 
neering at the Rensselaer Polytechnic Institute. Although 
those lectures, as given, form the basis of the work, they have^ 
of course, been considerably elaborated and extended, so as to 
cover many details of the subject which it would be impossible 
to include in any ordinary technical course of study, but which, 
at the same time, are necessary to a complete and philosoph- 
ical treatment. 

The first, or " Rational,'* part of this work, is intended to 
furnish an analytical or rational basis for the " Technical" or 
practical development contained in Part II. It will undoubt- 
edly impress a great number, and perhaps all engineers in 
active practice, that it is unnecessary to the proper treatment 
of such a subject. Indeed, a very considerably extended 
experience in iron and steel constructions places the author 
himself in position to fully appreciate the weight of such a 
criticism at the first glance. But it may be contended, and he 
thinks must be admitted, that the present advanced state of 
engineering as a profession implies the existence of something 
that may be called the " natural philosophy " of engineering. 
In other words, the engineer of the present time must meet 
the increased and increasing demands upon him in some 
one or more specialty, not only by the aid of sound common 



IV PREFACE. 

sense and a well-trained judgment, but also by a systematic 
knowledge of so much of natural philosophy as is involved 
in practical engineering operations. The ideal simplicity of 
stresses and strains in a perfectly isotropic body, and the clear- 
ness of action of *' external forces " applied at any " point " or 
distributed over any *' surface " according to some known and 
well-defined law, are not, it is evident, the things the technical 
student will encounter in his practice as an engineer. He will 
find fev\-. if any, of the ideal conditions realized, and the dififi- 
culties constantly confronting him will be those involving 
modifications of the analytical or mathematical results based 
upon ideal quantities and conditions. Nevertheless, it is cer- 
tainly true that in engineering practice he deals with precisely 
the same quantities as in the natural philosophy of engineering, 
but in different amounts and with far different and vastly more 
complicated conditions. And it is equally true that a correct 
knowledge of the consequent modifications, both in kind and 
amount must be based not only upon a correct recognition of 
the actual circumstances into which the ideal conditions trans- 
mute themselves in engineering works, /.r., upon sound prac- 
tical knowledge, but also upon a thorough comprehension of 
the things involved, in the abstract, and the lav/s governing 
their actions and relations. In other words, but in essentials 
the same, an engineer's preparation for active practice must 
consist both of that philosophical training in what is largely 
ideal, and which he acquires in the technical school, and of the 
purely practical training of the first few years of his profes- 
sional life. 

The first, or " Rational," part of this work is, then, designed 
for few others than technical students, although there are 
engineers whose tastes induce or circumstances require inves- 
tigations in connection with the elasticity and resistance of 
materials. The writer would esteem himself fortunate if the 
mathematical portion of the book should find favor with such 
individuals and be useful to them. 



PREFACE. V 

In Part II. the mathematical results obtained in Part I. are 
subjected to the test of experiment. By the aid of experi- 
mental results in a great variety of material, empirical coeffi- 
cients are established which involve the varied and complicated 
circumstances of material in actual use. The formulae, which 
otherwise express ideal conditions only, are thus rendered of 
the greatest practical value ; in fact they constitute the only 
reliable practical formulas in use by engineers. 

All the experimental results are, of course, compilations 
only, but they have been taken in all cases from what are 
believed to be trustworthy sources, and it has been the inten- 
tion to give credit to the experimenter in every case. It may 
appear that too great a profusion of experimental results has 
been introduced. But it has been the aim of the author, even 
at the risk of being tedious, to represent truly and completely 
the great variety of both quantitative and qualitative phenom- 
ena exhibited by material under test : to show not only the 
variation in products of different mills but the variation in 
different products of the same mill; to exhibit the variations 
due to difference in size, shape, relative dimensions and condi- 
tion of specimens : to show that specimens apparently identi- 
cally the same may even give considerable diversity in results 
and to prove the difference between the finished member and 
its component parts, as well as to indicate the direction in 
which further investigations may most profitably be prose- 
cuted. A few groups of tests are not sufficient to the attain- 
ment of such a series of results. 

In the course of the preparation of the ^ISS. the author 
found it necessary to reduce a ver}- great amount of experi- 
mental quantities from the crude shape of a mere record of 
tests to a useful condition, and to change many others from 
one unit to another. These numerical operations involved 
much labor, and although they were performed with great care 
and repeated in almost every instance, it is very probable that 
errors have crept in, though it is believed that there are few, 



VI PRE FA CE. 

if any, of importance. The writer will feel indebted to any 
one who will discover them. In all cases, unless otherwise 
specifically stated, the ultimate resistance, elastic limit and 
coefficient of elasticity are expressed in pounds per square 
inch of original area of section. 

In a few of the tables of Art. 32 the " strains," i.e., amounts 
of stretch, are given as decimal fractions (hundredths) of orig- 
inal length, while the otherwise uniform method of expression 
is by means of whole numbers giving per cents, of original 
dimensions. This diversity is unintentional and due to the 
fact that a part of the MSS. was a portion of that used in 
lectures. 

The distinction between " stress " and "■ strain " conflicts, 
so far as the latter word Is concerned, with ordinary usage. 
But some distinction is absolutely necessary, and that used has 
had a long existence, and is at least consistent with the ety- 
mology of the words. There certainly can be no way of 
filling the hiatus caused by the absence of a word to concisely 
express changes of shape or dimensions, without some Incon- 
venience, and that followed will probably cause as little as 
any. 

W. H. B. 

Rensselaer Polytechnic Institute, 

1883. 



CONTENTS. 



PART I.— RATIONAL. 
CHAPTER I. 

GENERAL THEORY OF ELASTICITY IN AMORPHOUS SOLID 

BODIES. 

ART. PAGE 

I. — General Statements i 

2. — Coefficients of Elasticity 3 

3, — Lateral Strains 4 

4. — Relation between the Coefficients of Elasticity for Shearing and Direct 

Stress in a Homogeneous Body 6 

5. — Expressions for Tangential and Direct Stresses in Terms of the Rates of 

Strains at any Point of a Homogeneous Body 8 

6. — General Equations of Internal Motion and Equilibrium 14 

7. — Equations of Motion and Equilibrium in Semi-Polar Co-ordinates 20 

8. — Equations of Motion and Equilibrium in Polar Co-ordinates 27 

CHAPTER n. 

THICK, HOLLOW CYLINDERS AND SPHERES, ANP TORSION. 

9. — Thick, Hollow Cylinders 36 

10. — Torsion in Equilibrium 43 

Equations of condition in rectangular co-ordinates 50 

Solutions of Equations (13) and (21) 53 

Elliptical section about its centre 54 

Equilateral triangle about its centre of gravity 57 



via CONTENTS. 



ART. PAGE 

Rectangular section about an axis passing through its centre of 

gravity 59 

Circular section about its ceiitre 75 

General observations 77 

II. — Torsional Oscillations of Circular Cylinders 78 

12. — Thick, Hollow Spheres 84 



CHAPTER III. 

THE ENERGY OF ELASTICITY. 

13. — "Work Expended in Producing Strains qo 

14. — Resilience 96 

15. — Suddenly Applied External Forces or Loads 98 

16. — Longitudinal Oscillations of a Straight Bar of Uniform Section 100 



CHAPTER IV. 

THEORY OF FLEXURE. 

17. — General Formulas 106 

18. — The Common Theory of Flexure 122 

19. — Deflection by the Common Theory of Flexure 130 

20. — External Bending Moments and Shears in General 132 

21. — Moments and Shears in Special Cases 138 

Case I 138 

Case II 139 

Case III 142 

22. — Recapitulation of the General Formulae of the Common Theory of 

Flexure I43 

23. — The Theorem of Three Moments 146 

23a. — Reactions under Continuous Beam of any Number of Spans 157 

Special method for ordinary use • • 105 

Special case of equal spans 169 

24. — The Neutral Curve for Special Cases 17^^ 

Case 1 172 

Case II 174 

Case III 177 



CONTENTS. IX 



ART. PAGE 

25. — The Flexure of Long Columns igo 

26. — Graphical Determination of the Resistance of a Beam ig6 

27. — The Common Theory of Flexure with Unequal Values of Coefficients of 

Elasticity 199 

28. — Greatest Stresses at any Point in a Beam 202 



PART II.— TECHNICAL. 
CHAPTER V. 

TENSION. 

29. — General Observations. — Limit of Elasticity , 208 

30. — Ultimate Resistance 210 

31. — Ductility. — Permanent Set 211 

32. — Wrought Iron 212 

Coefficient of elasticity 212 

Ultitnate resistance and elastic lifnit 223 

Wrought-iron boiler plate 241 

Effect of annealing 245 

Effect of hardening on the tensile resistance of iron and steel. . . 247 
Vafiation of tensile resistance ivith increase of temperature . ... 247 

Effect of low temperatures on wrought iron 251 

Iron wire 254 

Tensile resistance of shape iron 257 

English wrought iron 258 

Fracture of zvrought iron 258 

Crystallization of wrought iron 259 

Elevation of ultimate resistance and elastic limit 260 

Bauschinger's experiments on the change of elastic limit and 

coefficient 262 

Resistance of bar iron to suddenly applied stress 269 

Reduction of resistance betxveen the ultimate and breaking points. 269 

Effects of chemical constitution 270 

Kirkaldf s conclusions 272 

33. — Cast Iron 276 

Coefficient of elasticity and elastic limit 276 

Ultimate resistance 279 

Effect of rcmelting 282 



CONTENTS. 



ART. PAGE 

Effect of continued fusion 284 

Effect of repetition of stress 284 

Effect of high temperatures 286 

34.— Steel 286 

Coefficient of elasticity 286 

Ultimate resistance and elastic limit 294 

Boiler plate 306 

Effects of hardening arid tempering steel plates 314 

Rivet steel 315 

Effect of reduction of sectional area, in connection with ham- 

viering and rolling 315 

Effects of annealing steel 319 

Steel wire 319 

Shape steel 321 

Steel gun wire 322 

Effect of low and high temperatures on steel 323 

Effect of manipulations cojumon to constructive processes ; also 

punched, drilled and reamed holes 325 

Bauschinger's experiments on the change of clastic limit and 

coefficient of elasticity 333 

Fracture of steel 334 

Effect of chemical composition 334 

35. — Copper, Tin, Zinc and their Alloys. — Phosphor Bronze 336 

Coefficients of elasticity 336 

Ultimate resistance and elastic limit 338 

Gun metal 341 

General results 342 

Phosphor bronze, and brass and copper wire 344 

Experiments on rolled copper by the ^* Franklin Institute Com- 
mittee " 345 

Variation of ultimate resistance and stretch at high tempera- 
tures 345 

Bauschinger's experiments with copper and red brass 348 

36. — Various Metals and Glass 351 

Coefficients of elasticity 351 

Ultimate resistance and elastic limit 352 

37. — Cement, Cement Mortars, etc. — Brick 353 

Experiments and conclusions of Wm. W. Maclay, C. E 357 

Artificial stones 363 

Bricks 364 

Adhesion between bricks and cement mortar 364 

38.— Timber 365 



CONTENTS. XI 



CHAPTER VI. 

COMPRESSION. 

ART. PAGE 

39. — Preliminary 371 

40. — Wrought Iron ; 372 

41. — Cast Iron 376 

42. — Steel 380 

43. — Copper, Tin, Zinc and their Alloys 386 

44. — Glass 389 

45. — Cement. — Cement Mortar. — Concrete. — Artificial Stones 390 

46. — Brick 396 

47. — Natural Building Stones 398 

48. — Timber 403 



CHAPTER Vn. 

COMPRESSION. — LONG COLUMNS. 

49. — Preliminaiy Matter 409 

50. — Gordon's Formula 430 

51- — Experiments on Phoenix Columns, Latticed Channel Columns and Chan- 
nels 449 

Latticed channel columns and channels 455 

52. — Euler's and Tredgold's Forms of Long Column Formuloe 463 

53- — Hodgkinson's Formulae 469 

54* — Graphical Representation of Results of Long Column Experiments 473 

55- — Limit of Applicability of Euler's Formula 477 

56. — Reduction of Columns at Ends 479 

57. — Timber Columns 480 

Formula of C. Shaler Smith, C. E 485 



CHAPTER VIII. 

SHEARING AND TORSION. 

58. — Coefficient of Elasticity 487 

59. — Ultimate Resistance 490 

Wrought iron 49I 

Cast iron , 493 



XI 1 CONTENTS. 



ART. PAGE 

Steel 493 

Copper 495 

Timber 496 

60. — Torsion 498 

Coefficients of elasticity 498 

Ultimate resistance and elastic limit 498 

Wrought iron 498 

Cast iron 500 

Steel 504 

Copper^ tin, zinc and their alloys 507 

Timber 509 

Relation between ultitnate resistances to tension and torsion 510 



CHAPTER IX. 

BENDING, OR FLEXURE. 

61. — Coefficient of Elasticity 512 

62. — Formula for Rupture 512 

63. — Solid Rectangular and Circular Beams 514 

Wrought iron 515 

Cast iron 518 

Steel 520 

Cojubined steel and iron 523 

Copper, tin, zinc and their alloys 524 

Timber beams 526 

Timber beams of natural and prepared wood 536 

Cement, 7nortar and concrete 537 

Stone beams 543 

Practical for jnjilcE for solid beams 543 

Comparison of modulus of rupture for bending with ultimate 

resistances 545 

64. — Flanged Beams with Unequal Flanges 546 

Equal coefficients of elasticity 547 

Unequal coefficients of elasticity 550 

Cast-iron flanged beams 554 

Deflection of cast-iron flanged beams 558 

Wrought-iron X beams 559 

Deflection of wrought-iron X beams 562 

65. — Flanged Beams with Equal Flanges 564 

Experi7)ients of U. S. Test Board 575 



CONTENTS. xiii 



ART. PAGE 

66. — Built Flanged Beams Avith Equal Flanges. — Cover Plates 578 

67. — Built Flanged Beams with Equal Flanges. — No Cover Plates 589 

68. — Box Beams 5(^5 

69. — Exact Formulae for Built Beams 596 

70. — Examples of Built Beams Broken by Centre Weight 598 

Example I. — Wrought-iron beam .- 598 

Examp/e II. — Steel beam 600 

71. — Loss of Metal at Rivet Holes 601 

72. — Thickness of Web Plate 601 



CHAPTER X. 

CONNECTIONS. 

73. — Riveted Joints 606 

Kinds of joints 606 

Distribution of stress in riveted joints 607 

Effect of punching 615 

Wrought-iron lap joints^ and butt joints with single butt strap . 616 

Steel lap joints, and butt joints with ojte cover 627 

Wrought-iron butt joints with double covers 632 

Steel butt joints with double cover plates , 635 

Efficiencies . . 638 

Riveted truss joints 640 

Diagonal joints 642 

Friction of riveted joints 642 

Hand and 7nachine riveting 643 

74. — Welded Joints 643 

75. — Pin Connection 644 

76. — Iron, Steel and Hemp Cables or Ropes. — Wrought-iron Chain Cables. . . 64S 

Wrought-iron chain cables 652 



CHAPTER XI. 

MISCELLANEOUS PROBLEMS. 

77- — Resistance of Flues to Collapse 655 

78- — Approximate Treatment of Solid Metallic Rollers 659 

79- — Resistance to Driving and Drawing Spikes 663 

80. — Shearing Resistance of Timber behind Bolt or Mortise Holes 664 

81.— Bulging of Plates 665 



XIV CONTENTS. 



ART. PAGE 

82. — Special Cases of Flexure 674 

Flexure by oblique foxes 674 

General Jlexure by continuous ^ normal load 679 



CHAPTER XII. 

WORKING STRESSES AND SAFETY FACTORS. 

83. — Definitions 681 

84. — Specifications for Sabula Bridge 682 

85. — Specifications for Albany and Greenbush Bridge 686 

86. — Niagara Suspension Bridge 688 

87. — Menomonee Draw Bridge 689 

88. — Franklin Square Bridge 694 

89. — General Specifications 699 

90. — New York, Chicago and St. Louis Railway Specifications 699 

91. — Plattsmouth Bridge. 701 

92. — Specifications for Steel Cable Wire for the East River Suspension Bridge. 703 
93. — Specifications for Steel Wire Ropes for the Over-Floor Stays and Storm 

Cables of the East River Suspension Bridge 7^5 

94. — Specifications for Steel Suspenders, Connecting Rods, Stirrups and Pins, 

for the East River Suspension Bridge 7^5 

95. — Specifications for Certain Steel Work . . . East River Bridge, 

1881 706 



CHAPTER XIII. 

THE FATIGUE OF METALS. 

96.— Wohler's Law 708 

97. — Experimental Results 709 

98. — Formulae of Launhardt and Weyrauch 7^5 

99. — Influence of Time on Strains 7^9 

CHAPTER XIV. 

THE FLOW OF SOLIDS. 

100. — General Statements 723 

loi. — Tresca's Hypotheses 7^5 



CONTENTS. XV 



ART. PAGE 

102. — The Variable Meridian Section of the Primitive Central Cylinder 727 

103. — Positions in the Jet of Horizontal Sections of the Primitive Central 

Cylinder 729 

104. — Final Radius of a Horizontal Section of the Primitive Central Cylinder. 731 

105. — Path of any Molecule 731 



ADDENDA. 

To Art. 20 734 

To Art. 73 737 

To Art. 75 737 



Elasticity and Resistance of 
Materials. 



PART I.— rational. 



chapter I. 



General Theory of Elasticity in Amorphous Solid 

Bodies. 



Art. I. — General Statements. 

The molecules of all solid bodies known in nature are more 
or less free to move toward, or from, or among each other. 
Resistances are offered to such motions, which vary according 
to the circumstances under which they take place, and the 
nature of the body. This property of resistance is termed the 
" elasticity " of the body. 

The summation of the displacements of the molecules of a 
body, for a given point, is called the " distortioii '* or ^'' stram " 
at the point considered. ^\\^ force by which the molecules of 
a body resist a strain, at any point, is called the '■'■stress " at 
that point. This distinction between stress and strain is fun- 
damental and important. 

Stresses are developed, and strains caused, by the applica- 
tion of force to the exterior surface of the material. These 
stresses and strains vary in character according to the method 



2 ELASTICITY IN AMORPHOUS SOLID BODIES. [Art. I. 

of application of the external forces. Each stress, however, is 
accompanied by its own characteristic strain and no other. 
Thus, there are shearing stresses and shearing strains, tensile 
stresses and tensile strains, compressive stresses and compres- 
sive strains. Usually a number of different stresses with their 
corresponding strains are coexistent at any point in a body 
subjected to the action of external forces. 

It is a matter of experience that strains always vary con- 
tinuously and in the same direction with the corresponding 
stresses. Consequently the stresses are continuously increasing 
functions of the strains, and any stress may be represented 
by a series composed of the ascending powers (commencing 
with the first) of the strains multiplied by proper coefficients. 
When, as is usually the case, the displacements are very small, 
the terms of the series whose indices are greater than unity 
are exceedingly small compared with the first term, whose 
index is unity. Those terms may consequently be omitted 
without essentially changing the value of the expression. 
Hence follows what is ordinarily termed Hooke's law : 

The ratio between stresses and corresponding strains^ for a 
given material, is constant. 

This law is susceptible of very simple algebraic representa- 
tion. As the generality of the equation will not be affected, 
intensities of stresses and distortions or strains per linear unit, 
only, will be considered. 

Let /' represent the intensity of any stress, and /' the strain 
per unit of length, or, in other words, the rate of strain. If 
E' is a constant coefficient, Hooke's law will be given by the 
following equation : 

P' = ET (I) 

If the intensity of stress varies from point to point of a body, 
Hooke's law may be expressed by the following differential 
equation : 



Art. 2.] COEFFICIENTS OF ELASTICITY. 3 

-dT -^ (^) 

If/' and /' are rectangular co-ordinates, Eqs. (i) and (2) are 
evidently the equations of a straight line passing through the 
origin of co-ordinates. It will hereafter be seen that the line 
under consideration is essentially straight for very small strains 
only. 

Art. 2. — Coefficients of Elasticity. 

In general, the coefficient E' in Eq. (i) of the preceding 
Art., is called the '' coefficient of elasticity," or, sometimes, 
** modulus of elasticity." The coefficient of elasticity varies 
both with the kind of material and kind of stress. It simply 
expresses the ratio between stress arid strain. 

The characteristic strain of a tensile stress is evidently an 
increase of the linear dimensions of the body in the direction 
of action of the external forces. 

Let this increase per unit of length be represented by /, 
while/ and E represent, respectively, the corresponding in- 
tensity and coefficient. Eq. (i) of the preceding Art. then 
becomes : 

p = El, or, £ = | (I) 

E IS then the coefficient of elasticity for tension. 

The characteristic strain for a compressive stress is evi- 
dently a decrease in the linear dimensions of the body in the 
direction of action of the external forces. Let l^ represent this 
decrease per unit of length, p^ the intensity of compressive 
stress, and E^ the corresponding coefficient. Hence : 

/, =r EJ,, or, ^, = y (2) 



ELASTICITY IN AMORPHOUS SOLID BODIES. [Art. 3. 



E^, consequently, is the coefficient of elasticity for compres- 
sion. 

The characteristic strain for a shearing stress may be deter- 
mined by considering the effect which it produces on the layers 
of the body parallel to its plane of action. 

In Fig. I let ABCD represent one face of a cube, another 
of whose faces is fixed along AD, If a shear acts in the face 
BC^ whose plane is normal to the plane of the 
paper, all layers of the cube parallel to the 
plane of the shearing stress, i.e.^ BCy will slide 
over each other, so that the faces AB and DC 
will take the positions AE and DE. The 
amount of distortion or strain per unit of length 
will be represented by the angle EAB = cp. If 
the strain is small there may be written ^, sm cp 
or tan cp indifferently. 
Representing, therefore, the intensity of shear, coefficient 
and strain by 5, G and q)^ respectively, Eq. (i) of Art. i be- 
comes : 




Fig.l 



5 = Gcp^ or, G = 



9 



(3) 



It will be seen hereafter that there are certain limits of 
stress within which Eqs. (i), (2) and (3) are essentially true, 
but beyond which they do not hold ; this limit is called the 
" limit of elasticity," and is not in general a well defined 
point. 



Art. 3. — Lateral Strains. 

If a body, like that shown in Fig. i, be subjected to ten- 
sion, all of its oblique cross sections, such as EE and GH, will 
sustain shearing stresses in consequence of the components 
of the tension tangential to those oblique sections. These 



Art. 3.] 



LATERAL STRALNS. 



5 



tangential stresses will cause the oblique sections, in both 
directions, to slide over each other. Consequently tJie normal 
cross sections of the body will be decreased ; and if the normal 




Fig.l 



cross sections of the body are made less, its capacity of resist- 
ance to the external forces acting on AB and CD will be cor- 
respondingly diminished. 

If the body is subjected to compression, oblique sections of 
the body will be subjected to shears, but in directions opposite 
to those existing in the previous case. The effect of such 
shears will be an increase of the lateral dimensions of the body 
and a corresponding increase in its capacity of resistance. 

These changes in the lateral dimensions of the body are 
termed *' lateral strains"; they always accompany direct strains 
of tension and compression. 

It is to be observed that l^ateral strains decrease 3. body's 
resistance to tension, but increase its resistance to compression. 
Also, that if they are prevented, both kinds of resistance are 
increased. 

Consider a cube, each of whose edges is a, in a body sub- 
jected to tension. Let r represent the ratio between the 
lateral and direct strains, and let it be supposed to be the same 
in all directions. If /, as in Art. 2, represents the direct strain, 
the edges of the cube will become, by the tension : a{\ -\- /), 
a(\ — r/) and a{i — rl\ Consequently the volume of the re- 
sulting parallelopiped will be : 



a\\ + /) (i - rlj -^ a\\ + /(i - 2r)] 



(0 



6 ELASTICITY IN AMORPHOUS SOLID BODIES. [Art. 4. 

if powers of / higher than the first be omitted. With r be- 
tween o and ^, there will be an increase of volume, but not 
otherwise. 

If the body is subjected to compression, the edges of the 
cube become: <^(i — /j), a(\ -\- rj^ and a(\ -\- rj^\ while the 
volume of the parallelopiped takes the value : 

^3(1 _ /^) (I + rj^f = a\i + ll2r, - i)] . . . (2) 

As before, the higher powers of /^ are omitted. If the vol- 
ume of the cube is decreased, ^j must be found between o 
and 5^. 



Art. 4. — Relation between the Coefficients of Elasticity for Shearing and 
Direct Stress in a Homogeneous Body. 

A body is said to be homogeneous when its elasticity, of a 
given kind, is the same in all directions. 

Let Fig. I represent a body subjected to tension parallel to 
CD. That oblique section on which the shear has the greatest 
y^ E B intensity will make an angle of 

45° with either of those faces 
whose traces are CD or BD ; for 
if « is the angle which any 
oblique section makes with BD, 
■q P the total tension on BD, and 
A' the area of the latter surface, 
the total shear on any section whose area is A^ sec a, will be 
P sin a. Hence the intensity of shear is : 




P sin a P 

-r, = -IT, stn a cos a. 

A sec a A 



(0 



The second member of Eq. (i) evidently has its greatest 



Art. 4.] SHEARING AND DIRECT STRESS. 7 

value for a = 45°. Hence, if the tensile intensity on BD is 

p 
represented by -^ =/, the greatest intensity of shear will be : 

S = l (2) 

Then by Eq. (3) of Art. 2 : 

• ^-A • • • (3) 

In Fig. I EK and KG are perpendicular to each other, while 
they make angles of 45° with either AB or CD. After stress, 
the cube EKGH is distorted to the oblique parallelopiped 
E'KG'H'. Consequently EKGH and EKGH' correspond to 
ABCD and AEFD, respectively, of Fig. i. Art. 2. The angu- 
lar difference EKG — EKG' is then equal to cp ; and EKE' 

= GKG' = ^. Also EKE' = 45° _ ^. 

2 -^2 

Using, then, the notation of the preceding Arts., there will 
result, nearly : 

^«;. (45° - f ) = -^^ = I - /(I + r) ; . . (4) 

remembering that FK— FK(\ + /) ; and that 

E'F' = FK{i - rl). 

From a trigonometrical formula, there is obtained, very 
nearly : 

a CD CD 

. . tan 45 — tan -^ i — — 

^-(45°-f) = — : ^ = — i='-^- • « 

tan 45° -\- tan — i + — 
2 2 



8 ELASTICITY IN AMORPHOUS SOLID BODIES. [Art. 5. 

From Eqs. (4) and (5) : 

^ = /(i + ^) (6) 

Substituting from Eq. (3), as well as from Eq. (i) of Art. 2 : 

E 
G=--^ (7) 

2(1 -\- r) ^^ ^ 

It has already been seen in the preceding Art. that r must 
be found between o and ^, consequently the coefficient of elas- 
ticity for shearing lies between the values of y^ and J^ of that of 
the coefficient of elasticity for tension. 

This result is approximately verified by experiment. 

Since precisely the same form of result is obtained by 
treating compressive stress, instead of tensile, there will be 
found, by equating the two values of Gv 



It is clear, from the conditions assumed and operations 
involved, that the relations shown by Eqs. (7) and (8) can only 
be approximate. 



Art. 5. — Expressions for Tangential and Direct Stresses in Terms of the 
Rates of Strains at any point of a Homogeneous Body. 

Let any portion of material, perfectly homogeneous, be 
subjected to any state of stress whatever. At any point as O^ 
Fig. I, let there be assumed any three rectangular co-ordinate 
planes ; then consider any small rectangular parallelopiped 
whose faces are parallel to those planes. Finally let the 
stresses on the three faces nearest the origin be resolved into 



Art. 5.] 



STJ?£SSES IN TERMS OF STRAINS. 



components normal and parallel to their planes of action, 
whose directions are parallel to the co-ordinate axis. 

The intensities of these tangential and normal components 
will be represented in the usual manner, i.c,^ p^y signifies a 
tangential intensity on 
a plane normal to the 
axis of X (plane ^F), 
whose direction is paral- 
lel to the axis of F, 
while pxx signifies the 
intensity of a normal 
stress on a plane nor- 
mal to the axis of X 
(plane ZY^ and in the 
direction of the axis of 
X. Two unlike sub- 
scripts, therefore, indi- 
cate a tangential stress, while two of the same kind signify a 
normal stress. 

From Eq. (3) of Art. 2 and Eq. (7) of Art. 4, there is at 
once deduced : 




Fig.l 



5 = 



E 



2(1 + r) 



q) — Gq) 



(I) 



Now when the material is subjected to stress the lines 
bounding the faces of the parallelepiped will no longer be at 
right angles to each other. It has already been shown in Art. 
2 that the angular changes of the lines, from right angles, are 
the characteristic shearing strains, which, multiplied by G, give 
the shearing intensities. 

Let 9?j be the change of angle of the boundary lines 
parallel to X and Y. 

Let ^2 be the change of angle of the boundary lines 
parallel to Fand Z. 



10 ELASTICITY IN AMORPHOUS SOLID BODIES. [Art. 5. 

Let ^3 be the change of angle of the boundary lines 
parallel to Z and X, 

Eq. (i) will then give the following three equations : 

^- = W^) '^- •••••• (2) 

A«=^(I^)«'3 (4) 

In Fig. I let the rectangle agfh represent the right pro- 
jection of the indefinitely small parallelopiped dx dy dz. If 
M^ V and zv are the strains, parallel to the axis of x, y and z, of 

the original point //, the rates of variation of strain -7-, -^, -^-, 

dx dy dz 

etc., may be considered constant throughout this parallelo- 
piped ; consequently the rectangular faces will change to 
oblique parallelograms. The oblique parallelogram dJick^ whose 
diagonals may or may not coincide with those of agfJi^ there- 
fore, may represent the strained condition of the latter figure. 

Then, by Art. 2, the difference between /2%r and the right 
angle at h will represent the strain cp^. But, from Fig. i, 9?, 
has the following value : 

cp^ — dhe -f- bhc (5) 

But the limiting values of the angles in the second member 
are coincident with their tangents; hence: 

de , he f^ 

'^' = ^+^ ....... (6) 



Art. 5.] STJ?jSSSES in terms of strains. II 

But, again, de is the distortion parallel to OX fomtd by 
moving parallel to OY, only ; hence it is a partial differential of 
2/, or, it has the value : 

'^^ = f ^-^ (7) 

In precisely the same manner be is the partial differential 
of V in respect to x, or : 

, dv , 
oc = -~r dx, 
dx 

By the aid of these considerations, Eq. (6) takes the form : 

du , dv 

^■ = ^+z^ («) 

If XY be changed to XZy and then to ZX^ there may be 
at once written by the aid of Eq. (8) : 

dv dw . . 

^= = ^+^ ....... (9) 

dw du 
ax a 1^ 

Eqs. (2), (3) and (4) now take the following form : 



12 ELASTICITY IN AMORPHOUS SOLID BODIES. [Art. 5. 

The direct stresses are next to be given in terms of the 
displacements u^ v and w. Again, let the rectangular parallelo- 
piped dx dy dz be considered. Eq. (i), of Art. i, shows that 
the strain per unit of length is found by dividing the intensity 
of stress by the coefficient of elasticity, if a single stress only 
exists. But in the present instance, any state of stress what- 
ever is supposed. Consequently the strain caused by/^^,, for 
example, acting alone must be combined with the lateral 
strains induced hy />yy and/^j^. Denoting the actual rates of 
strain along the axes of X, V and Z by 4, 4 and l^ therefore, 
the following equations may be at once written by the aid of 
the principles given in Art. 3 : 



''.+ (a, +a. )|- (14) 



Ax _ T , f J. , ^ \ ^ 

E 



4^ = /.+ (a«+a. )^ (15) 



^=h+(p«+Px.)^ J16) 



P 3 ' \ ryy 1 Jr^xx 1 p 

Eliminating between these three equations : 

~A+Y^(^. + 4+/3)]- • • (17) 

'4 + Y^ (/. + /.+ 4)] . . . (18) 
3 + 7:^//. + 4 + 4)] • . . (19) 



E 

Pxx = -— , 



P: 



I -\- r 



Pzz = 



I + r 
But if u, V and w are the actual strains at the point where 



Art. 5] STRESSES I.V TERMS OF STRAINS. 1 3 

these stresses exist, the rates of strain /j, 4 and l^ will cvi- 

1.11 1 du dv . dzv . , rr^, 

dently be equal to - -, - - and -.- , respectively. 1 he volume 

dx dy dz 

of the parallelopiped will be changed by those strains to 
dxil + /.yXi + 4)^^<i + /a) = dx dy dz{i 4-/^4-4 + 4), 

if powers of 4, 4 and 4 above the first be omitted. The 
quantity (4 + 4 + 4) is, then, i]ie rate of variation of volume ^ or 
the amount of variation of volume for a cubic unit. If there be 
put 

,, du , dv , dzu . ^ E 

U =z -J -\- ^ — \- -J-, and G = 



dx ' dy ' dz ' 2(1 + ry 

Eqs. (17), (18) and (19) will take the forms : 

^Gr ^ ^ ^ du f . 



-Gr ^ , ^dv 

■ e -\- 2G — 

I — 2r dy 



Pyy=- ^^+26-— (21) 



A. = e ^ 2G ^ (22) 

^ \ — 2r dz 

The form in which Eqs. (14), (15) and (16) are written, 
shows that \i p^^, pyy or /^^, is positive, the stress is tension, 
and compression if it is negative. Consequently a positive 
value for any of the intensities in Eqs. (20), (21) or (22) will in- 
dicate a tensile stress, while a negative value will show the 
stress to be compressive. 

The Eqs. (14) to (19), together with the elimination in- 
volved, also show that the coefficients of elasticity for tension 



14 ELASTICITY IN AMORPHOUS SOLID BODIES. [Art. 6. 

and compression have been taken equal to each other, and that 
the ratio r is the same for tensile and compressive strains. 

Further, in Eqs. (ii), (12) and (13), it has been assumed 
that G is the same for all planes. 

Hence Eqs. (11), (12), (13), (20), (21) and (22) apply only to 
bodies perfectly homogeneous in all directions. 

It is to be observed that the co-ordinate axes have been 
taken perfectly arbitrarily. 



Art. 6. — General Equations of Internal Motion and Equilibrium. 

In establishing the general equations of motion and equi- 
librium, the principles of dynamics and statics are to be applied 
to the forces which act upon the parallelopiped represented in 
Fig. I, the edges of which are dx, dy and dz. The notation to 
be used for the intensities of the stresses acting on the dif- 
ferent faces will be the same as that used in the preceding 
Article. 

Let the stresses which act on the faces nearest the origin 
be considered negative, while those which act on the other 
three faces are taken as positive. 

The stresses which act in the direction of the axis of X are 
the following : 



On the face normal to JT, nearest to O 
" '' '' farthest from O 

" " dy dx nearest to O 

" '' " farthest from O 

" ** dz dx,nea.rGst to O 

" " '■' farthest from O 



—pxx dy dz. 

(^P.. -^ ^fdx^y dz, 

— pzxdy dx, 

[p.x+^fdz)dydx. 

— Pyx dz dx. 



Art. 6.] EQ UA TIOiVS IN RECTANGULAR CO-ORDINA TES. 



15 




The differential coefficients of the intensities are the rates 
of variation of those intensities for each unit of the variable, 
which, multiplied by the dif- 
ferentials of the variables, 
give the amounts of varia- 
tion for the different edges 
of the parallelopiped. 

Let Xq be the external 
force acting in the direction 
of X on a unit of volume at 
the point considered ; then 
Xq dx dy- d::: will be the 
amount of external force 
acting on the parallelopiped. 

These constitute all the forces acting on the parallelo- 
piped inthe direction of the axis of X, and their sum, if un- 

d^u 
balanced, must be equal to 7n —^dxdydz\ in which m is the 

mass or inertia of a unit of volume, and dt the differential of 
the time. Forming such an equation, therefore, and dropping 
the common factor dx dy dz, there will result : 




Fig.l 



dpxx , ^Pyx , ^Pzx \ V — .,, ^^^^ 



. . (I) 



Changing x to y, y to s, and z to x, Eq. (i) will become : 



% + % + ^. + F„ = ;« ^ . . . . (2) 
dx dy dz df- 



Again, in Eq. (i), changing ;ir to ^, ^ to y^ and y \.o x \ 



1 6 ELASTICITY IN AMORPHOUS SOLID BODIES, [Art. 6. 

The line of action of the resultant of all the forces which 
act on the indefinitely small parallelopiped, at its limit, passes 
through its centre of gravity, consequently it is subjected to the 
action of no unbalanced moment. The parallelopiped, therefore, 
can have no rotation about an axis passing through its centre 
of gravity, whether it be in motion or equilibrium. Hence, let 
an axis passing through its centre of gravity and parallel to the 
axis of Xy be considered. The only stresses, which, from their 
direction can possibly have moments about that axis, are those 
with the subscripts (/^), {py\ {^yy)', or (^-s-). But those with the 
last two subscripts act directly through the centre of the paral- 
lelopiped, consequently their moments are zero. The stresses 

-^~dy . dx dz and —^ds . dx dy are two of six forces whose 
ay dz 

resultant is directly opposed to the resultant of those three 
forces which represent the increase of the intensities of the 
normal, or direct, stresses on three of the faces of the parallelo- 
piped ; these, therefore, have no moments about the assumed 
axis. The only stresses remaining are those whose intensities 
are/^^ and/^^. The resultant moment, which must be equal 
to zero, then, has the following value : 

py^dx dz , dy -\- p^ydx dy , dz = o . . . . (4) 

••• A^= -P^y ••••••••.• (5) 

Hence the two intensities are equal to each other. 

The negative sign in Eq. (5) simply indicates that their 

moments have opposite signs or directions ; consequently, that 

the shears themselves, on adjacent faces, act toward or from 

the edge between those faces. In Eqs. (i), (2) and (3), the 



Art. 6.] EQ UA TIONS IN RECTANGULAR CO-ORDINA TES. 1 7 

tangential stresses, or shears, are all to be affected by the same 
sign, since direct, or normal, stresses only can have different 
signs. 

The Eq. (5) is perfectly general, hence there may be 
written : 

Pxy —pyx, and pzx^Pxz (6) 

Adopting the notation of Lam^, there may be written : 

A. = ^i, Pyy = N,, P„ = A^3, 

jPzy -^ iJ Pxz ^2, Pxy -^ 3> 

by which Eqs. (i), (2) and (3) take the following forms : 

^^ + ^+-^f +^° = '"^^7^- • • • (7) 

dT\ dN d1\ /V 

dx^ dy ^ ds ^ ^' dt' • • • • W 

dT, , dT. , dN, , _ d'w , , 

-di + ^+-d^+^' = '"-dT' • • • • (9) 

The equations (11), (12), (13), (20), (21) and (22) of the pre- 
ceding Art. are really kinematical in nature ; in order that the 
principles of dynamics may hold, they must satisfy Eqs. (7), (8) 
and (9). As the latter stand, by themselves, they are applica- 
ble to rigid bodies as well as elastic ones ; but when the values 
of N and 7", in terms of the strains u, v and w, have been in- 
serted they are restricted, in their use, to elastic bodies only. 
With those values so inserted, they form the equations on 
which are based the mathematical theory of sound and light 
vibrations, as well as those of elastic rods, membranes, etc. 

2 



1 8 ELASTICITY IN AMORPHOUS SOLID BODIES. [Art. 6. 

In general, they are the equations of motion which the dif- 
ferent parts of the body can have in reference to each other, 
in consequence of the elastic nature of the material of which 
the body is composed. 

If all parts of the body are in equilibrium under the action 
of the internal stresses, the rates of variation of the strains 

d^ii d^v d^iv 

-,— , -r- and — ^— , will each be equal to zero. Hence, Eqs. (7), 

(8) and (9) will take the forms : 

dNj^ . dT^ . dT^ , ^ . . 

^ + l5^ + -^7- + ^° = ° • • • • ('°) 

dx^ dy^ dz^ ^"-^ ' ■ ' ' ^"^ 
dT^ . dT^ . dNr, . ^ , . 

^ + l5r+-^^ + ^» = ° • • • • 02) 

These are the general equations of equilibrium. As they 
stand, they apply to a rigid body. For an elastic body, the 
values of N and T from the preceding Art., in terms of the 
strains u, v and w, must satisfy these equations. 

The Eqs. (10), (ii) and (12) express the three conditions of 
equilibrium that the sums of the forces acting on the small 
parallelopiped, taken in three rectangular co-ordinate direc- 
tions, must each be equal to zero. The other three conditions, 
indicating that the three component moments about the same 
co-ordinate axes must each be equal to zero, are fulfilled by 
Eqs. (5) and (6). The latter conditions really eliminate three 
of the nine unknown stresses. The remaining six conse- 
quently appear in both the equations of motion and equilib- 
rium. 

The equations (7) to (12), inclusive, belong to the interior 



Art. 6.] EQUA TIONS IN R EC TANG ULA R CO-OFDINA TES. 1 9 

of the body. At the exterior surface, only a portion of the 
small parallelopiped will exist, and that portion will be a 
tetrahedron, the base of which forms a part of the exterior 
surface of the body, and is acted upon by external forces. Let 
da be the area of the base of this tetrahedron, and let/, q and 
r be the angles which a normal to it forms with the three axes 
of Xy V and Zy respectively. Then will 

da cos p = dy ds, da cos q =^ ds dxy and da cos r = dx dy. 

Let P be the known intensity of the external force acting on 
day and let tt, x arid p be the angles which its direction makes 
with the co-ordinate axes. Then there will result : 

X^ = P da . cos Tty Fq — Pda . cos x and Z^ = Pda . cos p. 

The origin is now supposed to be so taken that the apex of 
the tetrahedron is located between it and the base ; hence that 
part of the parallelopiped in which acted the stresses involving 
the derivatives, or differential coefficients, is wanting ; con- 
sequently those stresses are also wanting. 

The sums of the forces, then, which act on the tetrahedron, 
in the co-ordinate directions, are the following : 

» 

— (iV, dy dz -{- T^ dz dx -\- T^dy dx) + Pda cos n = o^ 

— {T^dz dy -\- N^dz dx -\- T^dy dx) + Pda cos ^ = o, 

— {T^dz dy -\- T^dz dx + N^ dy dx) -\- Pda cos p = o. 
Substituting from above : 

iVj cos p -[- 7^3 cos q -\- T^ cos r = P cos n . . (13) 



20 ELASTICITY IN AMORPHOUS SOLID BODIES. [Art. J. 

T^ cos p -f- ^2 <^o^ ^ + ^1 <^os r — P cos X .. . (h) 
T^ cos p + Zi ^^j ^ + iV3 cos r = P cos ft ..(15) 

These equations must always be satisfied at the exterior 
surface of the body; and since the external forces must always 
be known, in order that a problem may be determinate, they 
will serve to determine constants which arise from the in- 
tegration of the general equations of motion and equilibrium. 



Art. 7. — Equations of Motion and Equilibrium in Semi-Polar 

Co-ordinates. 

For many purposes it is convenient to have the conditions 
of motion and equilibrium expressed in either semi-polar or 
polar co-ordinates ; the first form of such expression will be 
given in this Article. 

The general analytical method of transformation of co- 
ordinates may be applied to the equations of the preceding 
Article, but the direct treatment of an indefinitely small por- 
tion of the material, limited by co-ordinate surfaces, possesses 
many advantages. In Fig. i are shown both the small portion 
of material and the co-ordinates, semi-polar as well as rectangu- 
lar. The angle made by a plane normal to^F, and containing 
OXy with the plane XY is represented by cp ; the distance of 
any point from OX^ measured parallel to ZY^ is called r\ the 
third co-ordinate, normal to r and (p, is the co-ordinate x^ as 
before. It is important to observe that the co-ordinates ;r, r 
and <7?, at any point, are rectangular. 

The indefinitely small portion of material to be considered 
will, as shown in Fig. i, be limited by the edges dx, dr and 
r dcp. The faces dx'dr are inclined to each other at the angle 
d(p. 



Art. 7.] EQUATIONS IN SEMI-POLAR CO-ORDINATES. 21 

The intensities of the normal stresses in the directions of 
X and r will be indicated by N^ and R, respectively. The 
remainder of the notation 
will be of the same gen- 
eral character as that in 
the preceding Article ; 
i,e,, T^^ will represent a 
shear on the face dr . r dcp 
in the direction of r, while 
N^^ is a normal stress, in 
the direction of cp^ on the 
face dx dr. 

The strains or dis- 
placements, in the direc- 
tions of ,r, r and cp^ will 
be represented by Uy p 
and w ; consequently the 
unbalanced forces in those directions, per unit of mass, will be : 




d^u d^ft - d^w 
m -7—, m —J— and in — ^r- 
dt"" dt"" df" 



(I) 



Those forces acf ing on the faces hf, fe, and he, will be con- 
sidered negative ; those acting on the other faces, positive. 



Forces acting in the direction of r. 
— R . r dq) dx, and ; 

4- Rr dm dx 4- [ ^ , • dr = r —j- dr -\- R dr)dq) dx, 
\ dr dr J 



— T^ydr dx, and ; 



22 ELASTICITY IN AMORPHOUS SOLID BODIES. [Art. 7. 

-f- T^r^r dx ■\ — -^-- dcp, dr dx, 

— T^r . r dq) dr, and ; 

+ T^r . r dcp dr -\ -^ dx . r dq) dr. 

On the face dr dx, nearest to ZOX, there acts the normal 

stress ( N^^dr dx -\ -^ dcp, dr dx\— N'. Now N' has a com- 
ponent acting parallel to the face/^ and toward OX, equal to 
N' sin {dcp) z= iV'^l— ? = N'dcp, But the second term of this 

product will hold {dcpy, hence it will disappear, at the limit, in 
the first derivative of N'dcp .*. N'dcp — N^^dcp dr dx. Since 
this force must be taken as acting toward OX, it acts with the 
normal forces on Jif, and, consequently, must be given the 
negative sign. 

If Rq is the external force acting on a unit of volume, 
another force (external) acting along r will h^ R^ , r dcp dr dx. 

The sum of all these forces will be equal to 

m . r dcp dr dx . — ^— . 
^ de 



Forces acting in the direction of q), 

— N^4> dr dx, and ; 

4- N^^ dr dx -\ -—- dq) . dr dx. 

— Tr4> . r dq) dx, and ; 



Art. 7.] EQUATIONS IN SEMI-POLAR CO-ORDINATES. 23 

+ Tr^ . rdcpdx^i^^^^dr = r —^fdr + Tr^dr\dcp dx. 

— Tjc-i* • ^ dq) dry and ; 

+ 7*^,1, . r ^^ ^r -\ — -f^d^ ' ^ dq> dr. 

As in the case of N^^^ in connection with the forces along 
r, so the force T^rdr dx has a component along (^ (normal to 
/>) equal to T^^dr dx . sin idcp) = T^^dtpdrdx. It will have 
a positive sign, because it acts from OX. 

The external force is, 0^ . r dcp dr dx. 



Forces acting in the direction of x, 

— N^ . r dcp dr, and ; 

dN 
-\- N^r dcp dr -\ — -7—^ dx . r dcp dr, 

— Tyx • dx r dcp, and 

+ 7;, .dxrdcp-\- (^^"^ dr = r^^dr + Tr.dr\dxdcp, 

— 7(1,^ ^^' dr, and ; 

+ T^^dx dr + — r^ ^<?? . dx dr, 
dcp 

The external force is, X^ . r dcp dx dr. 

Putting each of these three sums equal to the proper rates 



24 ELASTICITY IN AMORPHOUS SOLID BODIES. [Art. /. 

oi variation of momentum, and dropping the common factor, 
r dcp dx dr\ 

dx ^ dr ^ r dcp ^ r ^ ° de ^ ^ 

dT^^ _^ dR _^ dT^r ^ R- N^<i> -{- Ro = m --^ (3) 



dx dr r dcp r dt^ 

-^r+ -w+ fw'^ — r — + ° " "^ ^^^ 

These are the general equations of motion (vibration) in 
terms of semi-polar co-ordinates ; if the second members are 
made equal to zero, they become equations of equilibrium. 
Eqs. (2), (3), and (4) are not dependent upon the nature of the 
body. 

Since x, r, and (p are rectangular, it at once follows that : 

^rx = ^xr> Tr^ = T^r, ^^id Tj^,j> = T^^, ... (5) 

In order that Eqs. (2), (3), and (4) may be restricted to 
elastic bodies, it is necessary to express the six intensities of 
stresses involved, in terms of the rates of variation of the strains 
in the rectangular co-ordinate directions of ;r, r, and cp. Since 
these co-ordinates are rectangular, the Eqs. (11), (12), (13), (20), 
(21), and (22) of Article 5, may be made applicable to the pres- 
ent case by some very simple changes dependent upon the 
nature of semi-polar co-ordinates. 

For the present purpose the strains in the co-ordinate direc- 
tions of ;tr, J/, and z will be represented by n! , v\ and w'. Since 
the axis of x remains the same in the two systems, evidently : 

du' _ du 
dx dx' 



Art. 7.] EQUATIONS ly SEMI-POLAR CO-ORDINATESr 2$ 

From Fig. i it is clear that the axis oi j/ corresponds exactly 
to the co-ordinate direction r ; hence : 

dv' dp 
dy dr 

From the same Fig. It is seen that the axis of z corresponds 

to cpy or rep. But the total differential, dw' , must be considered 

as made up of two parts ; consequently the rate of variation 

duo' 

' — ■ will consist of two parts also. If there is no distortion in 
dz 

the direction of r, or if the distance of a molecule from the 

onsfm remams the same, one part will be -r. — r = — — . if, 
^ d{rcp) rdcp 

however, a unit's length of material be removed from the dis- 
tance r to r -f P from the centre (9, Fig. i, while cp remains 

constant, its length will be changed from i to i . [ i -[- ~ )» i 

which p may be implicitly positive or negative. Consequently 
there will result : 

dw' _ dw p 
dz rd<p r 

For the reasons already given, there follow : 

du' _ dii J dv dp 

dy dr dx dx 

In Fig. 2 let dc be the side of a distorted small portion of 

the material, the original position of 

which was d'e. Od is the distance r ^~^^ 

from the origin, ad = dr and ac — dw, 

w^hile dd' = w. The angular change 

, y , ac dw . 
m position of dc is — , = -7—; but an 

ad dr 



in 




26 ELASTICITY IN AMORPHOUS SOLID BODIES. [Art. 7. 

amount equal to — . = — is due to the movement of r, and is 
ad r 

not a movement of </<^ relatively to the material immediately 

adjacent to d. 

Hence : 

dw' _ dw w , dv' __ dp 
dy dr r^ dz r dcp 



There only remain the following two, which may be at once 
written : 

dw' _ dw . dii _ dti 

dx dx dz r dcp 

The rate of variation of volume takes the following form in 
terms of the new co-ordinates : 



^ _ du' dv' dw' _du dp dw p ,^ 

dx dy dz ~. dx dr r dcp r ' ' ^ ' 



Accenting the intensities which belong to the rectangular 
system x, y, z, the Eqs. (11), (12), (13), (20), (21) and (22), of 
Art. 5, take the following form : 

N, =N^ = ^^^d-^2G^ (7) 

I — 2x dx ^^ 



R = N: = —^d^2G'^ (8) 

I — 2x dr ^ ^ 



iV^. = iV^3-^|^^+<^ + ^) ... (9) 



Art. 8.] EQUATIONS IN POLAR CO-ORDINATES. 2/ 

^-=^3' = <S+£) ao) 



^-=^-K5+;^) (-) 

If these values are Introduced in Eqs. (2), (3) and (4), those 
equations will be restricted in application to bodies of homo- 
geneous elasticity only. 

The notation r is used to indicate that the r involved is the 
ratio of lateral to direct strain, and that it has no relation 
whatever to the co-ordinate r. 

The limiting equations of condition, (13), (14) and (15) of 
Art. 6, remain the same, except for the changes of notation, 
shown in Eqs. (7) to (12), for the intensities N and 71 



Art. 8. — Equations of Motion and Equilibrium in Polar Co-ordinates. 

The relation, in space, existing between the polar and 
rectangular systems of co-ordinates is shown in Fig. i. The 
angle ^ is measured in the plane ZF and from that o( XV; 
while ip is measured normal to ZV in a plane which contains 
OX. The analytical relation existing between the two systems 
is, then, the following : 

X = r sin tpy y = r cos ip cos cp^ and s = r cos tp sin (p. 

The indefinitely small portion of material to be considered 
leaked. It is limited by the co-ordinate planes located by 



28 



ELASTICITY TV AMORPHOUS SOLID BODIES. [Art. 8. 



q) and f, and concentric spherical surfaces with radii r and 
r -j- dr. The directions r, (p and ^', at any point, are rectangu- 
lar; hence, the sums of the forces acting on the small portion 
of the material, taken in these directions, must be found and 
put equal to 



m 



~dP' 



m 



d^ 
'dt- 



V 



and 



in 



d^GD 

~di^' 



in which expressions, p, rj and od represent the strains in the 
direction of r, q) and ^' respectively. 




Those forces which act on the faces aJi, bd and cd will be 
considered negative, and those which act on the other faces 
positive. 

The notation will remain the same as in the preceding Ar- 
ticles, except that the three normal stresses will be indicated 
by N^, a; and N^. 



Art. 8.] EQUATIONS IN POLAR CO-ORDINATES. 2g 

Forces acting along r. 

— Nr , r dtp r cos ?/- dq), 

-{- Nr , r^ cos tp dip dcp + (^^^r^ dr = r'^ dr -\- 2r Nr dr\ 

cos ip dip dcp. 

— T^r • ^ dip dr, 

+ T^r . r dip dr -j j^ dq) . r dip dr, 

dcp 

— T^r • ^ <^os ip dcp dr. 

+ T^r ' r costp dcp dr -\- ( -^ — 5^^^- — ^ <///? = cos ip -— tt^ ^^ 

— T^r sin ip dipjr dq) dr. 

— N^ , r dip dr . si^i aOc = — N^ . r dtp dr . cos ip dcp ; 

on face ce, 

— N^ . r'cos tp dcp dr . si7t aOb = — N^ . r cos ip dcp dr , dip\ 

on face be. 

Forces acting along cp. 
—• Tr^ . r cos tp dq) r dip. 

+ Tr^ . T-cosip dq) dip + (^-^^^^-^ r^-^ ^r+ 2r Tr^d^ 

cos Ip dip dcp. 



30 ELASTICITY IN AMORPHOUS SOLID BODIES. [Art. 8. 



— N^ . r dtp dr. 



dN^ 



-\- N^ , r dip dr -{■ -j^dcp r dip dr. 

— T^^ . r cos ip dq) dr. 

+ T^^cosip. rdcp ^;.+ ^'iZkf^) dip = costp ^ dtp 

— T^^ sin rpdipjr dcp dr, 

•\- T^r ^ dip dr . cos tp dcp ; on face ce. 

— T^^ r dip dr I sin akc = j = — T^^ r dip dr , sin tp dcp; 

on face ce. 

The lines ak and ck are drawn normal to Oc and Oa, 

Forces acting along tp, 

— T^ , r cos tp dcp . r dip, 

+ Tr^ r^ cos tp dcp dip + (^-^^^ cir = 7^ ^± ar + 2r Tr^ dt\ 

cos tp dip dcp, 

— T^xj, . r dip dr, 

+ 7'^^ r dip dr -{- -~t^ dcp , r dip dr, 

— N^ . r costp dq) dr. 



Art. 8.] EQUATIONS IN POLAR CO-ORDINATES. 3 1 

-\- N^ . r cos ip dcp dr -^ i ~^ — ^^ ~ ^^^ ^ ~T^ ^^ 

— N^ sin ^dipjr dcp dr, 

+ T^r • ^ ^os tp dcp dr . dip ; on face be, 

■\- N^ , r dip dr . sin akc = + ^<^ r dip dr . sin tp dcp ; on face ce. 

The volume of the indefinitely small portion of the ma- 
terial is (omitting second powers of indefinitely small quan- 
tities) : 

r cos rp dq) , r dip , dr = ^V; 

and its mass is m multiplied by this small volume. The latter 
may be made a common factor in each of the three sums to be 
taken. 

The external forces acting in the directions R, cp and tp will 
be represented by : 

R^/IV, ^^AV and WJV, 

respectively. 

Taking each of the three sums, already mentioned, and 
dropping the common factor A V, there will result : 

dJVr dT^r , dT^r , 2Nr - N^ - N^ - T^r tau tp 

dr r cos tp . dq) r dip r 

+ ^»=-5r (I) 

dTr^ dN^ dT^^ 

dr r cos tp . dcp r dip 

2 Tr<}, + T^r — T^,}> tan tp — Ta^ tan tp , _ d^n , . 

r df" ^ ^ 



32 ELASTICITY IN AMORPHOUS SOLID BODIES. [Art. 8. 

dTr^ dT^^ dN^ 

dr r cos tp dcp r dip 

2 Tr^ + T^<\.r - N^ tan ^ -\- N^ tan tp , yr^ _ d^ , n 
Since r, q) and ip are rectangular at any point : 

Hence : 

2 r^^ + T^r — tan i-{ T^^ -\- T^^) _ 3 Tr4, — 2 tan tp . T^^ 



2Tr^ + T^r — tan ^'(A^^ — N^) _ 3 Tr^ — /^;^ ^'(iV^ — N^ ) 

These relations somewhat simplify the first members of 
Eqs. (2) and (3). 

Eqs. (i), (2) and (3) are entirely independent of the nature 
of the material ; also, they apply to the case of equilibrium, if 
the second members are made equal to zero. 

The rectangular rates of strain, at any point, in terms of 
r, cp and ?/j are next to be found. As in the preceding Art, 
the rates of strain in the rectangular directions of r, q) and ^ 
will be indicated by : 

dv dw' du dv du' 

W d^' d^' d^" dy" ^^^* 

Remembering the reasoning in connection with the value of 
nr— , in the preceding Art., and attentively considering Fig. i, 
there may at once be written ; 



Art. 8.] EQUATIONS IN POLAR CO-ORDINATES. 33 

dii! _ doD p 

dx r dip r' 

In Fig. \/\{ ac — \ and ab = go, while ak = r cot .ip (ak is 
perpendicular to aO\ the difference in length between ac and 
bh will be : 

Qo GO tan 4' 



r cot tp 



This expression is negative because a decrease in length takes 
place in consequence of a movement in i\iQ positive direction 
of rip. 

Again, a consideration of Fig. i, and the reasoning con- 
nected with the equation above, will give : 

dw' _ drf p 00 tan ^ 

dz' r cos tp dcp r r 

Without explanation there may at once be written : 

dv _ dp 
dy' dr ' 

Fig. I of this, and Fig. 2 of the preceding Art. give : 

dti' dco GO , dv' dp 

1 = -TT , and - 



dy dr r ' dx r dip' 

These are to be used in the expression for T^r^ Precisely 
the same Figs, and method give : 

dv' __ dp _. dw' _ drf tj ^ 

dz r cos ip dcp ' dj/' dr r ' 

which are to be used in finding T^r* 

3 



34 



ELASTICITY IN AMORPHOUS SOLID BODIES. [Art. 8. 



The expression for -^—7 will be composed of the sum of two 

parts. In Fig. 2, ab is the original position of r dip, and after 
the strain rf exists it takes the position ec. Consequently ac 

(equal and parallel to bd and perpendicular 
to ak) represents the strain 7;, while cd rep- 
resents dq. Since, also, fc is perpendicu- 
lar to ck, the strains of the kind 7/ change 
the right angle fck to the angle fee ; or 
the angle eck is equal to 

dw' J , 1 1 ^d ea 

—rr = eed -\- dek r= — - _i- _- 
dx de ' ak 



/— 




= -J- 4- ^ 
r dip r eot tp 



In Fig. 2, the points a, b and k are iden- 
tical with the points similarly lettered in Fig. i. The expres- 

du 
sion for ■— may be at once written from Fig. i. There may, 

CI' M 

then, finally be written : , 



dw' 
dx 



drf Tf tan ip 



r dip 



. du doD 

and, -r-, = r— 

dz r eos ip dq) 



These equations will give the expression for T^^, 
The value of 



^ _ die' dv' dw' 
dx dy dz' 



now takes the following form : 

drf 



dr r eos rp dcp 



dcD , 2p GO tan . . 

r dip r r - 



Art. 8.] EQUATIONS IN POLAR CO-ORDINATES. 35 

The last two terms are characteristic of the spherical co- 
ordinates. 

The equations (20), (21), (22), (11), (12) and (13), of Art. (5), 
take the forms : 

N. =^^e + 2g'^4 (5) 

N, = ^^ e + 2g( '^y , + ^ - 2L^'^^\ (6) 

I — 2r \r cos ^ dcp r r J ^ ^ 

N,=^^e + 2G(^+!^) (7) 

"^ I — 2r \r dip rj 

7;,= c(^+ _i^+^i^). ... (8) 

\r dip r cos ^ dcp r J 

„ ^fdoD GO dp \ , X 



T^=g(^-^+^JI-^) (10) 

\r cos f dcp dr rJ 

If these values are inserted In Eqs. (i), (2) and (3), the 
resulting equations will be applicable to isotropic material 
only. 

As in the preceding Art., r is used to express the ratio 
between direct and lateral strains, and has no relation what- 
ever to the co-ordinate r. 

It is interesting and important to observe that the equa- 
tions of motion and equilibrium for elastic bodies, are only 
special cases of equations which are entirely independent of 
the nature of the material, of equations, in fact, which express 
the most general conditions of motion or equilibrium. 



CHAPTER II. 



Thick, Hollow Cylinders and Spheres, and Torsion. 

Art. 9. — Thick, Hollow Cylinders. 



In Fig. I is represented a section, taken normal to its axis, 
of a circular cylinder whose walls are of the appreciable thick- 
ness /. Let/ and/i represent the interior and exterior inten- 
sities of pressures, respectively. The material will not be 
stressed with uniform intensity throughout the thickness /. Yet 

if that thickness, comparatively speak- 
ing, is small, the variation will also be 
small ; or, in other words, the intensity 
of stress throughout the thickness t 
may be considered constant. This 
approximate case will first be con- 
sidered. 

The interior intensity / will be 

considered greater than the exterior 

/i, consequently the tendency will 

'^' be toward rupture along a diametral 

plane. If, at the same time, the ends of the cylinder are taken 

as closed, as will be done, a tendency to rupture through the 

section shown in the Fig. will exist. 

The force tending to produce rupture of the latter kind 
will be : 




F = n{pr'^ - p,r^) 



(I) 



Art. 9-] THICK, HOLLOW CYLINDERS. 37 

If iVj represents the intensity of stress developed by this 
force, 

N = . ^ = />r'^-Ar.' .... (2) 

If the exterior pressure is zero, and if r is nearly equal to 



^ - 2{r- r)~ 2t ^^^ 

In this same approximate case, the tendency to split the 
cylinder along a diametral plane, for unit of length, will be : 

F' = pr' - /,r,. 

If N' is the intensity of stress developed by F' : 

^'^^^Pr'-P^r, (4) 

N' is thus seen to be twice as great as Nj_ when/^ = o. If, 
therefore, the material has the same ultimate resistance in both 
directions the cylinder will fail longitudinally when the interior 
intensity is only half great enough to produce transverse rup- 
ture ; the tJiickness being assumed to be very small and the ex- 
terior pressure zero, 

iVj and N' are tensile stresses, because the interior pressure 
was assumed to be large compared with the exterior. If the 
opposite assumption were made, they would be found to be 
compression, while the general forms would remain exactly the 
same. 



38 THICK, HOLLOW CYLINDERS. [Art. 9. 

The preceding formulas are too loosely approximate for 
many cases. The exact treatment requires the use of the 
general equations of equilibrium, and the forms which they take 
in Art. 7 are particularly convenient. As in that Art., the axis 
of X will be taken as the axis of the cylinder. 

Since all external pressure is uniform in intensity and nor- 
mal in direction, no shearing stresses will exist in the material 
of the cylinder. This condition is expressed in the notation 
of Art. 7 by putting : 

J- ^x — ^ rx — J- r<^ — 0» 

Again the cylinder will be considered closed at the ends, 
and the force F^ Eq. (i), will be assumed to develop a stress 
of uniform intensity throughout the transverse section shown 
in Fig. I. This condition, in fact, is involved in that of making 
all the tangential stresses equal to zero. 

Since this case is that of equilibrium, the equations (2), (3) 
and (4) of Art. 7 take the following form, after neglecting X^, 
Rq and (?o : 

dR R-N^^^ (,) 

dr r ^ ^ 

^ = o (7) 

These equations are next to be expressed in terms of the 
strains 2/, p and w. 

In consequence of the manner of application of the external 
forces, all movements of indefinitely small portions of the 



Art. 9.] THICK, HOLLOW CYLINDERS. 39 

material will be along the radii and axis of the cylinder. 
Hence : 

u will be independent of r and ^; 
p " " ♦* '' (p '' x\ 

w ■— o. 

The rate of change, therefore, of volume will be (Eq. (6) of 
Art. 7) : 

Q^du^dp^p^ (8) 

ax dr r 



dd d^'ii 
As p is independent of ;r, -7- = — — ; hence if the value of 

dx dx'^ 

Nj^ be taken from Eq. (7) of Art. 7 and put in Eq. (5) of this 

Art. : 

dN^ _ 2Gx d^ii ^d^iL __ 

dx I — 2r dx^ dx^ 

d^u , 

,\ — ,~ = O, and u = ax -j- a , 
dx^ 

But the transverse section in which the origin is located 
may be considered fixed. Consequently if ;ir = o, 21 = and 
thus a' = o. The expression for n is then : ?/ = ax. 

The ratio // -^ ;i: is the /' of Eq. (i), Art. i ; while the /' 
of the same equation is simply N^ of Eq. (2), given above. 
Hence : 

X E E{r^ — r") ^^^ 

Again, Eq. (8), of Art. 7, in connection with Eqs. (8) and (6) 
of this, gives : 



40 THICK, HOLLOW CYLINDERS. [Art. 9. 



2Gx fd^p _L_ ^ _ 8\ \ 2G(^ Jl. ^ ^ 

— 2x\dr'' r dr r^j \dr^ r dr r^J 



— o. 



d\^ 



d^p , dp p d^p , \r , 

• i_ _!_ i_ mr — -4- ■ -- nz Q. 

dr^ r dr r^ dr^ dr 



dp , p 
r 



■'■ ^ + r = ^;°r: 



r dp -\- p dr = d(pr) = cr dr. 



cr^ , , cr ^ b . . 

pr = \- d; or, p =— 4- — . . . (10) 



This value of p in Eqs. (8) and (9) of Art. 7 will give : 



Ji = 2G\'-i^^-±^+ £ -11 . . . (II) 

I I — 2r 2 r' \ ' 



AT,, = 2C i H(!i±i) + ^ + 4 I . . . (12) 
^^ [ I — 2r 2 r^ [ ^ 

At the interior surface R must be equal to the internal 
pressure, and at the exterior surface to the external pressure. 
Or, since negative signs indicate compression ; 

li r = r' R= -p, 

li r = r, i? = — /,. 

Either of these equations is the simple result of applying 
Eqs. (13), (14) and (15) to the present case, for which, 



Art. 9.] THICK, HOLLOW CYLINDERS. 4I 

COS p = COS r = cos it = cos p = o, 

cos q = cos X ^ \y and P =. — p or — p-v 

Applying Eq. (11) to the two surfaces: 

^ { x{a -\- c) ^ c b ) . . 

— / = 2G \ ^ M rA • • • (13) 



-A-^r^^+T-^4- • • (H) 



Subtracting (14) from (13) : 

2Gb = (A - / ) ^' ^'' 



.'2 ^ 2 

I 



Inserting this value in Eq. (13): 



[ I — 2r 2 j 



/2 _ ^2 



The general expressions of R and A^,^,/,, freed from the arbi- 
trary constants of integration, can now be easily written by 
inserting these last two values in Eqs. (j i) and (12). By making 
the insertions there will result : 

^ ^ A^;x^ - pr' _ (A - /) ry^ . 1 . . . he) 



The stress A^.^,^ is a tension directed around the cylinder, and 



42 THICK, HOLLOW CYLINDERS. [Art. 9. 

has been called '' hoop tension." Eq. (16) shows that the hoop 
tension will be greatest at the interior of the cylinder. An ex- 
pression for the thickness, /, of the annulus in terms of the 
greatest hoop tension (which will be called //) can easily be 
obtained from Eq. (16). 

\i r ^^ r' in that equation : 



h 



_ 2/,r,^ - / {r'^ + r^) 






h-Vp 



r' \2A - / -f /^ 

■•■'■■- ^ ='= 'i (s^^J- ■ i •■ <-> 

Eq. (17) will enable the thickness to be so determined that 
the hoop tension shall not exceed any assigned limit //. If /i 
is so small in comparison with/ that it maybe neglected, / 
will become : 



If/i is greater than /, N^^ becomes compression, but the 
equations are in no manner changed. 

The values of the constants b and c may easily be found 
from the two equations immediately preceding Eq. (15). 

It is interesting to notice that the rate of change of volume, 
By is equal to {a -j- ^) and, therefore, constant for all points. 



Art. 10.] 



TORSION IN EQUILIBRIUM. 



43 



Art. 10. — Torsion in Equilibrium. 



The formulas to be deduced in this Article are those first 
given by Saint-Venaut, but, with one or two exceptions, es- 
tablished in a different manner. 

It will in all cases, except that of the final result for a rec- 
tangular cross section, be convenient to use those equations of 
Art. 7 which are given in terms of semi-polar co-ordinates. 

Let Fig. I represent a cylindrical piece of material, with 
any cross section, fixed in the plane ZY^ and let the origin of 
co-ordinates be taken at O. Let 
it be twisted, also, by a couple 

P.ab = PI, 



the plane of which is parallel to 
ZY. The material will thus be 
subjected to no bending, but to 
pure torsion. 

The axis of the piece is sup- 
posed to be parallel to the axis of 
A^as well as the axis of the couple. 
Normal sections of the piece, orig- 
inally parallel to ZO Y, will not re- / P'S*' 
main plane after torsion takes place. But the tendency to 
twist any elementary portion of the piece about an axis pass- 
ing through its centre and parallel to the axis of Jf will be very 
small compared with the tendency to twist it about either the 
axis of r or ^ ; consequently the first will be neglected. In 
the notation of Art. /, this condition is equivalent to making 
Tr^ = o. 

As the piece is acted upon by a couple only, all normal 
stresses will be zero. 




44 TORSION IN EQUILIBRIUM. [Art. 10. 

Eqs. (7), (8), (9) and (i i), of Art. 7, then become : 

iV, = d -Y2G~- =0 (i) 

\ — 2X dx 



R =^^^+2C^ = o . (2) 

\ — 2X dr ^ ^ 



^^ 2GV n , r-f dw , P\ r . 



f dp . dw . w\ . , 



After introducing the values of T^^ and Zif,^, from Eqs. (10) 
and (12) of Art. 7, in Eqs. (2), (3) and (4) of the same Article, 
at the same time making the external forces and second mem- 
bers of those equations equal to zero, and bearing in mind the 
conditions given above, there will result : 



dT^x dT^x T^x _ ^f d^tc d^p d^w d^ti 

dr r dcp r \ dr^ dr dx r dqj dx r'dqy^ 

. du . dp \ , . 



dT^x ^[ d^'ii . d^p\ ,^. 



dx ~ W '^ r dcp dx) - o . . . . I7j 



Also by Eq. (6) of Art. 7 : 



Art. 10.] TORSION IN EQUILIBRIUM, 45 

e — -^i Jr-^ Jr — — 4- - fs^i 

dx dr r dcp r ^ ^ 

The cylindrical piece of material is supposed to be of such 
length, that the portion to which these equations apply is not 
affected by the manner of application of the couple. This 
portion is, therefore, twisted uniformly from end to end ; con- 
sequently the strain u will not vary with any change in x. 
Hence : 

du f V 

^ = ° fe) 



Eq. (i) then shows that = o. This was to be anticipated, 

since a pure shear cannot change the volume or density. Be- 
cause 6 = 0, Eqs. (2) and (3) at once give : 

dp dw , p , . 

~ = —1- + - = O (10) 

dr rdcp r ^ ^ 

As the torsion is uniform throughout the portion con- 
sidered : 

dp _ _ dp , . 

dx r dx ^ 

Eq. (11) in connection with Eq. (10), gives: 

d^w 



r dx dcp 



(12) 



Eqs. (11) and (12), in connection with Eq. (10), reduce Eq. 
(5) to the following form : 



46 TORSION IN EQUILIBRIUM. [Art. 10. 

d(r- 
d'u , d'u , du d'u , V dr/ , . 



r* dcp^ dr^ r dr dqf dr 

Both terms of the second member of Eq. (6) reduce to zero 
by Eqs. (9) and (11), and give no new condition. The second 
term of the second member of Eq. (7) is zero by Eq. (9) ; the 
remaining term therefore gives : 

d'^w f . 

d^=° ('4) 

As the stress is all shearing, p will not vary with cp. 
Hence : 

dp / V 

■^d^ = ° • • • ('5) 

Eqs. (10), (11) and (15) show that p = o, and reduce Eq. 
(4) to: 

dw w ■ / ^\ 

d^-v = ° • • • ('^) 

div 
Eq. (10) now becomes — -j- = o, and shows that w does not 

r d(p 

contain cp ; while Eq. (14) shows that iv does not contain x"" or 

any higher power of x. The strain w, in connection with 

these conditions, is to be so determined as to satisfy Eq. (16). 

If <a' is a constant, the following form fulfills all conditions : 

w = arx (17) 

Eq. (17) shows that the strain w, in the direction of q), i.e., 
the angular straiit at any pointy varies directly as the distance 



Art. lO.] TORSION IN EQUILIBRIUM. 4/ 

from the axis of X, and^ as the distance from the origin measured 
along that axis. This is a direct consequence of making Tr,^ 
= o. 

The quantity a is evidently the angle of torsion, or the 
angle through which one end of a unit of fibre, situated at 
unit's distance from the axis, is twisted ; for if ; 

r=;ir=l, w = a. 

An equation of condition relative to the exterior surface of 
the twisted piece yet remains to be determined ; and that is 
to be based on the supposition that no external force whatever 
acts on the outer surface of the piece. In Eqs. (13), (14) and 
(15) of Art. 6, consequently, P = o. The conditions of the 
problem also make all the stresses except : 

7^3 = T^r and 7^ = T^^ 

equal to zero, while the cylindrical character of the piece 
makes : 

/ = 90° .*. cos p = o. 

If cos t be written for cos r : 

cos t = sin q. 

Eq. (13), just cited, then gives : 

T^^ cos q -\- T^^sinq~o (18) 

But since p = o and w = arx: 

^-=4" ('9) 



48 



TORSION IN EQUILIBRIUM. ' 



[Art. 10. 



and 



Tx^ = G 



du 
r dcp 



+ (xr 



Eq. (i8) now becomes: 

du 
dr 



du 
r dcp 



tan q 



-\- ar 



^ .... (20) 



dr^ 
r^dcp ' 



. . . (21) 



in which r^ is the value of r for the perimeter of any normal 
section. 

Eqs. (13) and (21) are all that are necessary and all that 
exist, for the determination of the strain //. Eq. (13) must be 
fulfilled at all points in the interior of the twisted piece, while 
Eq. (21) must, at the same time, hold true at all points of the 
exterior surface. 

After u is determined, T^,^ and J^^ at once result from Eqs. 
(19) and (20). The resisting moment of torsion then becomes: 



M = 



du 



T^A r^ dcp . dr = G -j~ .r dr dcp -f Gah . (22) 

dq) '' 



In this equation 7^ = r^ . r dcp dr is the polar moment of 

inertia of the normal section of the piece about the axis of 
X, and the double integral is to be extended over the whole 
section. 

According to the old, or common, theory of torsion : 

M =• Gal^, 

The third member of Eq. (22), shows, however, that such an 
expression is not correct unless u is equal to zero, i.e.^ unless 
all normal sections remain plane while the piece is subjected 



Art. 10.] 



TORSION IN EQUILIBRIUM. 



49 



to torsion. It will be seen that this is true for a circular sec- 
tion only. 

It may sometimes be convenient to put Eq. (22) in the 
foUowincr form : 



J/= G 



' du f 

r dr . -J— dcp + Gal. = G ti , r dr -\- Gal. . (23) 
dcp J 



In this equation 71 is to be considered as : 



['^du , 



while the remaining integration in r Is to be so made that the 
whole section shall be covered. 

It is very important to observe that the equations of con- 
dition for the determination of ti, and consequently the general 
values of T^^ and T^^, are wholly independent of any consider- 
ations regarding the position of the axis of torsion, or the axis 
of X. It follows from this, that the resistance of pure torsion is 
precisely the same wherever may be the axis about which the 
piece is twisted. It is to be borne in mind, however, that, if 
the axis of the twisting is not the axis of the cylindrical piece, 
the latter will be subjected to combined bending and torsion ; 
the bending being produced by a force sufficient to cause the 
piece to take the helical position ne- 
cessitated by the torsion. The cylin- 
drical axis is the straight line locus of 
the centres of gravity of all the normal 
sections. 

If, as in Fig. 2, there are n cylinders 
whose centres c are all at the same 
distance Cc — I from the centre C of 
twisting, or motion ; and if M is the 
total moment of torsion of the system, 

4 




Fig.2 



50 TORSION IN EQUILIBRIUM. [Art. lO. 

while ffi is the moment of torsion of each cyHnder about its 
own axis or centre r, then will M = n7n ; and each cylinder 
will be subject to a bending moment whose amount can be 
determined from the condition that the diameter of each piece 
lying along Cc before torsion, must pass through C after, and 
during, torsion, also. 

Since T^^ and 7^^ act at right angles to each other, the re- 
sultant intensity of shear at any point in an originally normal 
section of the twisted piece will be : 



T=VTj-{- T^^^ (24) 

According to the ordinary methods of the calculus, the co- 
ordinates of the point at which T has its greatest value must 
satisfy the equations : 

dT dT f . 

d^T/ , d^ / / d^T \^ _d'T d^T / 
dcp^^'' dr' \°' \dcpdr) dqj' ' dr' ~^' 

After the solution of Eqs. (25), it will usually be necessary 
only to inspect the resulting value of T, in order to determine 
whether it is a maximum or minimum, without applying the 
tests which follow those equations. 



Equations of Condition in Rectangular Co-ordinates, 

In the case of a rectangular normal section, the analysis is 
somewhat simplified by taking some of the quantities used in 
terms of rectangular co-ordinates. 

In the notation of Art. 6, all stresses will be zero except 



Art. 10.] TORSION IN EQUILIBRIUM. 51 

7^3 and T^, Hence Eqs. (lo), (ii) and (12) of that Article re- 
duce to : 

dy dz 



III 

dx 



dT\ 
dx 



= o. 



= o. 



The strains in the directions of x, y and z are, respectively, 
u^ V and w. Introducing the values of T^ and T^ in the equa- 
tions above, in terms of these strains, from Eqs. (ii) and (13) 
of Art. 5 ; and then doing the same in reference to the con- 
ditions 

N^^ N, = N^^ 7; =: o : 

the following equations will result : 

d^'u , d^u , ^. 

dy+dF = ° (^^) 

dv , dw , . 

;& + <^ = ° • • •■ • • • • (^7) 

The operations by which these results are reached are iden- 
tical with those used above in connection with semi-polar co- 
ordinates, and need not be repeated. 

Eq. (27) is satisfied by taking : 

V = axz ; 
w = — axy ; 

in which a is the angle of torsion, as before. 



52 



TORSION IN EQUILIBRIUM. 



[Art. 10, 



Eqs. (ii) and (13) of Art. 5 then give : 



\dz dx J \dz 



. . . (28) 



. . (29) 



The element of a normal section is dz dy. Hence the mo- 
ment of torsion is 



M 



i< 



T^z — 7; j) dy dz. 



M= G 



jy'-Tz^^ ^yd'^^^h . (30). 



.-. M = G 



{zu dz — yu d}) -f- Gal^ . . . . (31) 



// = 



{z'-^-f) dy dz 



is the polar moment of inertia of any section about the axis 
ofX 

The integrals are to be extended over the whole section ; 
hence, in Eq. (31), zu dz is to be taken as : 



z dz . 



+ y, 



° du , 



and yu dy 2J^\ 



y dy 



r+^. 



^ du J 
-zdz 



Art. 10.] TORSION IN EQUIIIBRIUM. 53 

in which expressions, y^ and z^ are general co-ordinates of the 
perimeter of the normal section. 

Eq. (26) is identical with Eq. (13), and can be derived from 
it, through a change in the independent variables, by the aid 
of the relations: 

2 =z r cos cp ; and y ^ r sin cp. 



Solutions of Eqs. (13) and {21), 

It has been shown that the function u, which represents the 
strain parallel to the axis of the piece, must satisfy Eq. (13) [or 
Eq. (26)] for all points of any normal section, and Eq. (21) (or 
a corresponding one in rectangular co-ordinates) at all points 
of the perimeter ; and those two are the only conditions to be 
satisfied. 

It is shown by the ordinary operations of the calculus that 
an indefinite number of functions u, of r and cp, will satisfy Eq. 
(13); and, of these, that some are algebraic and some tran- 
scendental. 

It is further shown that the various functions tc which 
satisfy both Eqs. (13) and (21) differ only by constants. 

If ti is first supposed to be algebraic in character, and if c^^ 
^2, Cj^ etc., represent constant coefficients, the following general 
function will satisfy Eq. (13) : 

u = a\ ^'/ ^^^^ 9 ^ ^2^^ sin 2cp + c^ r^ sin ^cp -\- \ .) , . 

i -\- c\r cos q) -\- c'^r^ cos 2q) -f- ^3 t^ cos 3<p -{-.,) 

and the following equation, which is supposed to belong to the 
perimeter of a normal section only, will be found to satisfy 
Eq. (21): 

\- c^r cos cp -\- c^r^ cos 2q) -{- c^ r^ cos j^cp + • • 

— c-i^r sin cp — c^r'^ sin 2(p — c^ r^ sin ^cp — . , . = C (33) 



54 TORSION IN EQUILIBRIUM. [Art. 10. 

(7 is a constant which changes only with the form of sec- 
tion. 

If —r and — r- be found from Eq. (32), while - — ~ be taken 
dr r dcp r^ dcp 

from Eq. (33), and if these quantities be then introduced in Eq. 

(21), it will be found that that equation is satisfied. 

The only form of transcendental function needed, among 

those to which the integration of Eq. (13) or Eq. (26) leads, 

will be given in connection with the consideration of pieces 

with rectangular section, where it will be used. 



Elliptical Section about its Centre, 

Let a cylindrical piece of material with elliptical normal 
section be taken, and let a be the semi-major and b the semi- 
minor axis, while the angle q) is measured from a with the 
centre of the ellipse as the origin of co-ordinates, since the 
cylinder will be twisted about its own axis. The polar equa- 
tion of the elliptical perimeter may take the following shape : 

^2 ^2 Jf — a^ a^Jf 

h — • r cos 2q> = — . . . (34) 

2^2 a' -{- b' ^ a' -}- b' ^^^^ 

By a comparison of Eqs. (33) and (34), it is seen that : 

b^ ~ a^ , ^ a^b^ 

^2 = ,/ , , /,x ; and C = 



2{a^ + b') ' a' ^ b'' 

and that all the other constants are zero. Hence Eq. (32) 
gives : 

b^ — a^ . c^ 

u = Of . ^ , — ^ r^ sin 20) — — fr^ sin 2(p . . (35) 
2{a^ -^ b^) ^ 2 -^ ^ ^^^^ 

The quantity represented by /is evident. 



Art. lo.] 



TORSION IN EQUILIBRIUM. 



55 



By Eqs. (19) and (20) : 



b' - a' 



(36) 



'1)2 _ ^2 

r^^ = Ga[^^-—^^ r cos 2tp + r 



) . . . (37) 



T » 7' d(J) 

Since ° ' ° • = ^^, ^ being the area of the ellipse, or 

naby the second member of Eq. (22), by the aid of Eq. (37), 
may take the form : 



M ^ Ga 



^^lo(: 



a' 



, r^ cos 2q) + r^\dr, 
a^ ^ b^ -r -r y 



.-. J/ = 6^ 



a 



a^ -{- b^ 4 ^ ^ 4 J ^ 



Then using Eq. (34) : 



M= Ga 



a'b^ 



a" + b^] 



dA = Ga 



TTO^b^ 

a" 4- b^ 



. (38) 



If Ip is the polar moment of inertia of the ellipse (i.e.^ about 
an axis normal to its plane and passing through its centre), so 
that 

_ _ 7rab{a^ -\- b^) 



then : 



M = Ga 



_A^ 
4^% 



(39) 



5^ TORSION IN EQUILIBRIUM. [Art. lO. 

Using f in the manner shown in Eq. (35), the resultant 
shear at any point becomes, by Eq. (24) : 





T — Gar Vf 4- 2f cos 2cp -|- i. 




dT 


gives : 






sin 2^ = 0, or cp — 90° or 0°. 



Since / is negative, T will evidently take its maximum 
when (p has such a value that 2/ cos 2(p is positive ; or, q) must 
be 90°. 

Hence the greatest intensity of shear will be found some- 
where along the minor axis. But the preceding expression 
shows that T varies directly as the distance from the centre. 
Hence, the greatest intensity of shear is found at the extremities 
of the minor axis. 

Making cp = 90° and r = d in the value of T: 

2a^b 
T= T^= Gab{l -/) = G^-2~:^2 ' ' • (40) 

Taking Ga from Eq. (40) and inserting it in Eq. (38) : 

M^ T„^ = 2Tjf; (41) 

in which : 

J. _ Ttah^ 



or the moment of inertia of the section about the major axis. 



Art. 10.] 



TORSION IN EQUILIBRIUM. 



57 



Equilateral Triangle about its Centre of Gravity, 

This case is that of a cylindrical piece whose normal cross 
section is an equilateral triangle, and the torsion will be sup- 
posed about an axis passing through the 
centres of gravity of the different nor- 
mal sections. The cross section is rep- 
resented in Fig. 3, G being the centre 
of gravity as well as the origin of co- " 
ordinates. 

Let GH = }4GD = a. Then from 
the known properties of such a triangle - f 




Fig.3 



FD = BB = BF = 2a V3. 

2^ r cos CD 

Hence, the equation for DB is ; r sin cp — — _ ^ = o. 

Hence, the equation for BF is ; r cos cp -\- a =0. 

Hence, the equation for FD is ; r sin cp -[ — — l — o. 

Taking the product of these three equations, and reducing, 
there will result for the equation to the perimeter : 



r^ r^ 2a^ 

— — •^- cos '\q) =^ — 
2 6^ ^^ 3 



(42) 



Comparing this equation with Eq. (33): 



c^— — ^- \ and, 6 = — . 
6^ 3 



Hence: 



58 



TORSION IN EQUILIBRIUM, 



[Art. lO. 



u — — a 



r^ silt ^cp 
6a 



And by Eqs. (19) and (20) : 



Tjfrr = — G^ 



r^ sz7t ^cp 
2a 



Eq. (22) then gives : 



M = Gal^ - Ga 



r* cos '\(p J J 

'^^ dr dcp, 

2a 



(43) 



(44) 



^ r- f r"" COS 3(p\ . . 

T^^-=- Ga[r ^^j .... (45) 



= Galp — Ga 



r* sin xq) , 

^ ^^ dr. 

oa 



Gailp a'^^/ -yj = 0.6 Galp = 1.8 Ga a*^/^; (46) 



since I^ = polar moment of inertia == ^a^Vs- 
By Eq. (24) : 



a 



r* 

^a^ 



. . , . (47) 



dT 
,\ -~j — = o, gives sm 3^ = o ; 

or <?> r= 0°, 60°, 120°, 180°, 240°, 300° or 360°. 

The values 0°, 120°, 240° and 360° make : 

cos icp = -\- \\ 



Art. lO.] TORSION IN EQUILIBRIUM. 59 

hence, for a given valoie of r, these make T a minimum. The 
values 60°, 180° and 300° make: 

cos ^cp — — \\ 

hence, for a given value of r these make T a maximum. 
Putting cos icp = — \ in Eq. (47) : 

r=(;«(r + J) (48) 

This value will be the greatest possible when r is the 
greatest. But qy = 60°, 180° and 300°, correspond to the nor- 
mal a dropped on each of the three sides of the triangle from 
G, Hence r = a, in Eq. (48), gives the greatest intensity of 
shear T^, or : 

^m = 7 Gaa (49) 

Or, the greatest intensity of shear exists at the middle point of 
each side. Those points are the nearest of all, in the perimeter, 
to the axis of torsion. 

The value of Ga, from Eq. (49), inserted in Eq. (46), gives : 

M=OA-'T„=~; (50) 

a 20 

in which /= side of section =: 2a V3. 



Rectangular Section about an Axis passing through its Centre 

of Gravity. 

In this case it will be necessary to consider one of the 
transcendental forms to which the integration of Eq. (13) [or 



6o TORSION IN EQUILIBRIUM. [Art. 10. 

(26)] leads ; for if the polar equation to the perimeter be formed, 
as was done in the preceding case, it will be found to contain 
r^ to which no term in Eq. (33) corresponds. 

If e is the base of the Napierian system of logarithms (nu- 
merically, e— 2.71828, nearly), and A any constant whatever, 
it is known that the general integral of the partial differential 
equation (13) may be expressed as follows : 

when n^ -j- n'^ — o. For : 

dr^ ^ r^ dcj? ' r dr ^ ^ ^ 

But the second member of this equation is evidently equal 
to zero if 



(71^ H- 72'^) = o, or n' ■= ^/ — 7f. 

These relations make it necessary that either n or n! shall be 
imaginary. 

It will hereafter be convenient to use the following notation 
for hyperbolic sines, cosines and tangents : 

^ — ^""^ C^ -4- C''^ €^ — €~^ 

sih t = ; cv/i t = ■ ; and, ta/i / = — ^ 



e' + e-' 



By the use of Euler's exponential formula, as is well known, 
and remembering that 71'^ = — 71^, Eq. (51) may be put in the 
following form : 

u = 2c'^^^^^<l> \_A„ S171 (7tr si 71 cp) -{- A'n cos {jir si7t cp)\ ; 
in which the sign of summation is to be extended to all pos- 



Art. 10.] TORSION IN EQUIIIBRIUM. 6l 

sible values of A^ and A'^. At the centre of any section for 
which r is zero, u must be zero also, for the axis of the piece is 
not shortened. This condition requires that ^^ — o ; u then 
becomes : 

\i — ^e'^^''"'"^ A„ sin {iir sin q)). 

The subsequent analysis will be simplified by introducing 
the form of the hyperbolic sine, and this may be done by 
adding and subtracting the same quantity to that already 
under the sign of summation, in such a manner that : 

u = ^[A„ sin {nr sin cp) . siJi (jir cos cp) 

■\-y2A^sin(nr sin cp) e-''''"'"^^ .... (52) 

. Now if the product : 

sin {nr sin q)) e~'"^'^''^'^ 

be developed in a series and multiplied by A„, one term will 
consist of the quantity : 

— r^ sin cp cos cp 

multiplied by a constant, and if: 

'^A^ sin (nr sin cp) e-'^'' ^'^ ^ 

be replaced by simply : 

— ar^ sin cp cos cp 

all the conditions of the problem will be found to be satisfied. 
This is equivalent to putting: 

— ar^ sin cp cos cp 



62 



TORSION IN EQUILIBRIUM. 



[Art. 10. 



for a general function of r sin cp and r cos cp. This change will 
give the following form to u^ first used by Saint- Venant : 

u = '^A^ sin (nr sin cp) . sih {nr cos cp) — ar^ sin cp cos cp . (53) 

« 
Fig. 4 represents the cross section with C as the origin of 

co-ordinates, and axis. The angle q) is measured positively 




from CN toward CH. At the points N, H, K and Z, in the 
equation to the perimeter, dr^ will be zero. Hence, at those 
points, by Eq. (21) : 

-r = '^\An ^^fi (^^^ si'^ (p) • ?^ cos cp . coh (nr cos cp) 

-\- An . n sin cp . cos {nr sin cp) . sih {nr cos cp)'] 

— 2ar sin cp cos cp = o. 

At the points under consideration cp has the values 0°, 90°, 
180°, 270° and 360°. At the points A^and K, cp = o" or 180° ; 

Cll£ 

hence, sin cp = o, and both terms of the second member of -— 

dr 

reduce to zero, whatever may be the value of n. But at H and 

Lj cp = 90° and 270° ; hence, sin cp = -\- i or — i and cos cp = o. 



In order then, that -,- 

dr 



Q ^.t H and Z, these must obtain : 



Art. lO.] TORSION IN EQUIIIBRIUM. 63 

COS nr = cos (— 7tr) = o. 

If HL = c; and, KN = b ; then : 



no f nc^. , , 

cos— = cos[-—)=o (54) 



If the signification of n be now somewhat changed so as to 
represent all possible whole numbers between o and 00, Eq. (54) 
will be satisfied by writing : 

2n — \ ^ 

• It 



for n, in that equation Eq. (53) will then become : 

" - . (2n — I • \ •/ /2;^ — I \ 

u = ^An Sin ( Ttr sin cp J . si/i f 7rr cos (p) 

— ar^ sin (p cos (p (55) 

The quantity A^ yet remains to be determined by the aid 
of Eq. (21), which expresses the condition existing at the 
perimeter of any section. 

Now, for the portion BN of the perimeter : 



r cos <z> = — , 

2 



and — -~ will be the tangent of (— 9) ; or, 

J— = — tan (— cp) = tan cp. 



Hence, Eq. (21) becomes: 



^4 TORSION IN EQUILIBRIUM. [Art. lO. 



du 
dr 



du 
r dip 



= tan <p (56) 



or 



du du 

ar stn g? = -— cos cp — sin a>, 

dr r dcp 



Substituting from Eq. (55), then making: 



r cos (7? = — : 
^ 2 



s° ^ 2n — I J f2n — I A 
r stn w = ^A^ . ■ tt . co/i I tto ) 

^ 2ac \ 2C J 



2n ~ \ 
. sm ( nr sin cp 



If r sin cp be represented by the rectangular co-ordinate 
jj/, and another quantity by //", the above equation may be 
written : 

y^H^ sin ^ + H^sin ?^ + H^sin ^^ 



^ c 



-\- , , , — H^ sin [ It \ y ^ ... 



If both sides of this equation be multiplied by 

7ty\ , dyy 



'271 — I 

sm 



Art. 10.] TOaSIOiV IN EQUILIBRIUM. 



65 



C . 



and if the integral then be taken between the Hmits o and -, it 

2 

is known from the integral calculus that all terms except the 
11^^ will disappear, and that : 



r 




"2 


n 




J 


c 


, 


"2 




C 


) 



211 I 

y . si/i \ ^- Tty ] . dy 



i/i- ( U^ . Try ) . dy. 



Completing the'se simple integrations : 



//. 



\i27l -\)7t) ^ '^ ' c' 



Hence 



A^ = 



(- ly-^c^ 4 



2ac 



(2n — If 71"" ' C ' {211 — l) 7T ' 



coil [ 7zb 



2C 



If this value of A,, be put in Eq. (55), and if rectangular co- 
ordinates : 

y z= r sill (}), and z ~ r cos cp, 
be introduced, that equation will become : 



u 



/2\3 



ac"^ 



(_ ,)«-., /«(?^ „y ) . ,i,, (^JLZI n, ) 



{211 - 1)3 coh i^'^:^ nb \ 



■ (57) 



66 



TORSION IN EQUILIBRIUM. 



[Art. 10. 



This value of u placed in Eq. (31) will enable the moment 
of torsion to be at once written. 

c c 

The limits -fjo and — y^ are -]- - and ; and the limits 

1 , <^ 1 <^ TT 

4- z^ and — z^ are ^ — and . Hence : 

' ° 2 2 



u 



=:a be 



.J f2n — I 

2 sill 7tZ 



{2n — 1)3 r<?/^ f — nb \ 



= Qy for brevity. 



u 



= abc 



c 



. , (—iY~''.2sihl- — Ttb ) . siji 

2\Zc ^ ^ \ 2C J 

7t) b 1 r N„ r f2n — I 



- TTJ j 



(2;2 — 1)3 coh 



2C 



nb 



= R 



For the next integration : 



Qz dz = abc 



12 



,(2\c ^ {2n-\)n 



2<^^ , 2n—\ J 
. r<?/2 Ttb 



2c {2n—lfn 



sih [ Ttb 



2C 



(2;/— 1)3 CO J I ( nb 



Art. 10.] TORSION IN EQUILIBRIUM. 



67 



2^ 


\'c ^ 


Ry dy = abc 

2 

, ^'' sih 

, {271 - If TT^ 


_I2 


nj 


' b ~ 


{271 — 1)3 coh { 


2C J 



Thus the Integrations Indicated in Eq. (31) are completed. 
Hence : 



M=g\ Qzdz-\- Ry dy -\- alA, 



Remembering that : 



L = be 



\2 



M=Ga 



'bc^ i6bc^ " I 

.6 71* ^ {271 —If 



^ , tah r~ ^- 7tb 

64c* ^_ \ 2C 

n^ I {2n — i)"^ 



• • (58) 



But it is known that : 



;t4 



^{27^ — ly I . 2 . 3 ' 25 * 



Hence Eq. (58) becomes : 



68 



TORSION IN EQUILIBRIUM, 



[Art. 10. 





I 


../. i^'"- ' -r^l 




M - Gabc^ 


_ 64 ^ ^, ^""^^ \ 2C ) 
7is b I {211 — 1)5 


• (59) 


Since: 










a 


+ ?+?-■■) 




(\ — tah 7T. 
\ I 


+ - 


I — tah 3;t , I — tah ^tt , 





tail 7t 
I 


+ 


tah ;^7r ^ tah ^tt ^^ 

3' 5' 


• > 


and since: 




-^ = 0.209137, 





and remembering that : 



:^ 



2;i- ij 3' S' 



7T^ 



2V 295.1215 



Eq. (59) becomes : 



M= Gabc^ 



0.210083 7- 

L3 .^ 



I — tah 
+ 0.209137 ^\ h 



I — tah ~ — 



2C 



+ 



(60) 



Eq. (60) gives the value of the moment of torsion of a rec- 



tangular bar of material. 



Art. 10.] 



TORSION IN EQUILIBRIUM. 



69 



If z had been taken parallel to b, and y parallel to r, a 
moment of equal value would have been found, which can be 
at once written from Eq. (60) by writing b for c and c for b. 

That moment will be : 



M = Gacb^ 



0.210083 - 

L3 c 



+ 0.209137 



b 



tall 



TTC 

Tb 



+ 



tah 



3^^ 

2b 



3^ 



+ 



(61) 



Eq. (60) should be used when b is greater tJian e, and Eq. (61) 
wJien c is greater than b^ because the series in the parentheses 
are then very rapidly converging, and not diverging. It will 
never be necessary to take more than three or four terms and 
one, only, will ordinarily be sufficient. The following are the 
values of, 



I — tah 



nn 



for a few values of n 



1 mt\ 
I — tah- — ) = 0.083 '• 0-00373 : 0.000162 : 0.000007. 

n — 1 \ 2 : X : a. 



Square Section, 
li c = b either Eq. (60) or Eq. (61) gives : 

M — Gab* 0.2101 4- o.209( I — tah - j 



70 TORSION IN EQUILIBRIUM. [Art. 10. 

,\ M ^ 0.1406 Gab^ = Ga ^\ ; {62) 

42.7 A 



in which A is the area ( = d^) and I^ is the polar moment of 



mertia ( = z- y 



Rectangle m which b = 2r. 
If <^ = 2<:, Eq. (60) gives : 

M — Ga . 2c^( 0.105 + 0.1046 (i — tah it) 

A^ 

.*. M = 0.457 ^^'^^^ == G^ r ' (^3) 

42^^ 

in which A is the area (= 2c^) and I^= polar moment of inertia 

_ bc^ 4- b^c _ Sf^ 
~ 12 ~ 6 ' 



Rectangle in which b = 4c, 
li b = 4c, Eq. (60) then gives : 

M = Gabc^ I 0.0525) = 1. 123 Gac^ 

■•■ ^^=^«7^; (64) 

40.2 I^ 
in which A = area = 4c' and / = polar moment of inertia 



Art. 10.] TORSION IN EQUILIBRIUM. J I 

~ 12 ~ 3 ' 

If b is greater than 2c, it will be sufficiently near for all 
ordinary purposes to write : 

M=gJ':Ui -0-63j) (65) 



Greatest hitensity of Shear. 

There yet remains to be determined the greatest intensity 
of shear at any point in a section, and in searching for this 
quantity it will be convenient to use Eqs. (28) and (29). 

It will also be well to observe that by changing ^ to j', y 
to — Zy e to b and b to e, in Eq. (57), there may be at once 
written : 

^ (- iy-'si7Z f ^^^ ~ ^ tt;: ^ . si/i C^^^-^ ^f) 

• ^^ ^ An - I \ (66) 

{2n - 1)3 eo/i [^ ^^ nc j 

This amounts to turning the co-ordinate axes 90°. 
Since the resultant shear at any point is : 



it will be necessary to seek the maximum of 



dii \ ^ fdu \ T^ 



72 TORSION IN EQUILIBRIUM. [Art. lO. 

The two following equations will then give the points de- 
sired : 



--(S 


7 fdu \ d'u 


dy 


- \dy ''- "") df 


/dn 
^ \dz 


"y)Udy-'') 


<S)_ 


fdu , \ / d^'u 
\dy + "^j \d. dy - 


dz 


+ ( 


fdu \ d^'u 

.d. - "V d:^ = ° 



) = o . . . (67) 



+ a 



. . (68) 



It is unnecessary to reproduce the complete substitutions in 
tnese two equations, but such operations show that the points 
of maximiun values of T are at the middle points of the sides of 
the rectangular sections ; omitting the evident fact that T—O 
at the centre. It will also be found that the greatest in- 
tensity of shear ivill exist at the middle points of the greater 
sides. 

This result may be reached independent of any analytical 
test, by bearing in mind that an elongated ellipse closely ap- 
proximates a rectangular section, and it has already been 
shown that the greatest intensity in an elliptical section is 
found at the extremities of the smaller axis. 

By the aid of Eqs. (28), (29), (57) and {66), it will also be 
found that ^^3 = at the extremities of the diameter c, and 
7^2 = o at the extremities of the diameter b. The maximum 
value of T will then be : 



Art. 10.] 



TORSION IN EQUIIIBRIUM. 



73 



T^= - 7\- - G i"-^ - ^J' 



. (69) 



By the use of Eq. (57) : 



du 



- S^jz+l^""' 



/ N« T ^ .211—1 \ J f 271 — I 

(— i)''"^ . — . stn ( ny j . coli ( — n^ 



ac"^ 



(271 —if coh I — Ttd 

^ ^ \ 2C 



Putting XT = o and y = — in this equation, there will result : 



ZL = Gac 



^ 



. . J I 271 — I , 

{271 — 1/ co/i (^ no 



■ (70) 



If b is greater than c the series appearing in this equation is 
very rapidly convergent, and it will never be necessary to use 
more than two or three terms if the section is square, and if b 
is four or five times c there may be written : 



T,„ = Gc>:c 



(70 



Square Section. 



Making b — c m Eq. (70) and making ;^ = i, 2 and 3 {t.e.y 
taking three terms of the series) there will result : 



74 TORSION IN EQUILIBRIUM. [Art. 10. 

T 

T,n = 0.676 Gac .'. Goi — 1.48--. 

Inserting this value in Eq. (62) : 

M =0,21 b^T,, = ^ - '^^a^^ .... (72) 

/. T,„=O.Z-j-a = S -^j ' • . . (73) 



in which : 



r ^^ ^ be 

/ == — and a = ~ = ~ . 
12 22 



Rectangular Scetion ; b = 2e. 

Making b = 2e in Eq. (70) and making ?i = i, only, there 
will result : 

T^ = 0.93 6^ar .*. Ga = 1.08 — ^ . 
Inserting this value in Eq. (6;^) : 

M = 0.49 c'T„,= 1,47 -'^ .... (74) 



T^ = 0.68 -^ ^ = 2 -— ; . . . . (75) 
J e^ 



in which : 



J. be^ t* , e 

J = — = -p- and a = — , 
126 2 



Art. lO.] TORSION IN EQUILIBRIUM. 75 

Rectangular Section ; b ■= 4c. 
Making b = 4c in Eq. (70) and making 7i = 1, only : 

T 
T^ = 0.997 Gac .*. Ga = 1.003 — ^« 

Inserting this value in Eq. (64) : 

M= 1.126 c^T„,= i.6g^^ .... (76) 

a ^ ^ 



.-. T^ z=z 0.6 -J- a = 0.9 --;.... {yy) 



in which : 



r bc^ c^ . c 

1 = — ■ = — and a = — 

12 ^ 2 



Circular Section about its Centre, 

» 

The torsion of a circular cylinder furnishes the simplest 
example of all. 

If r^ is the radius of the circular section, the polar equation 
of that section is : 



r" 

-^ = Cy (constant). 



Comparing this equation with Eq. (33), it is seen that 

C J c 2 1 2 ... — C J — C 2 — ... — - vJ. 



7^ TORSION IN EQUILIBRIUM. [Art. lO. 

By Eq. (32) this gives u = o. Hence, all sections remain 
plane during torsion. 

Eqs. (19) and (20) then give : 

T^r — O', and, Tx<^ = Gar .... (yS) 
Eq. (23) gives for the moment of torsion : 

M = Galp (79) 

or : 

M = 0.5 7t7^^ . Ga = -a (80) 

In which equation, A is the area of the section and 

T 'q 

^ 2 

The greatest intensity of shear in the section will be ob- 
tained by making r = r^in Eq. (78) ; or : 

T^ = Gar,^ .'. Ga='^ (81) 

Eq. (80) then becomes : 

M = o.s Ttrl T^ = 2^-^ (82) 



.-. T^ = 0.64 — - = 0.5 -- r, ; .... {S3) 
in which 

4 



Art. 10.] ^ TORSION IN EQUILIBRIUM. 77 

It is thus seen that the circular section is the only one 
treated which remains plane during torsion. 



General Observations. 

The preceding examples will sufficiently exemplify the 
method to be followed in any case. Some general conclusions, 
however, may be drawn from a consideration of Eq. (33). 

If the perimeter is symmetrical about the line from which 
cp is measured, then r must be the same for -f cp and — (p\ 
hence : 

c\ = e\ = Cj^ = , . . —o. 

If the perimeter is symmetrical about a line at right angles 
to the zero position of r, then r must be the same for : 

q) = 90° -j- cp' and 90° — (p' ; 
hence : 

e,= e^= e^ . . . = c, = e'^ = e'e = . . . = O. 

In connection with the first of these sets of results, Eq. (32) 
shows that everf axis of symmetry of sections represented by Eq. 
(33) will 7iot be moved from its original position by torsion. 

If the section has two axes of symmetry passing through 
the origin of co-ordinates, then will all the above constants be 
zero, and its equation will become : 



y2 

— 1- c^r^ cos 2(p -\- c^r^ cos 4^ -\- e^r^ cos 6q) -f . . . " = K, 



7^ TORSIONAL OSCILLATIONS. [Art. II. 



Art. II. — Torsional Oscillations of Circular Cylinders. 

Two cases of torsional oscillations will be considered, In the 
first of which the cylindrical body twisted is supposed to be 
the only one in motion. In the second case, however, the mass 
of the twisted body will be neglected, and the motion of a 
heavy body, attached to its free end, will be considered. In 
both cases the section of the cylinder will be considered cir- 
cular. 

Since these cases are those of motion, the internal stresses 
are not, in general, in equilibrium ; hence, equations of motion 
must be used, and those of Art. 7 are most convenient. Of 
these last, the investigations of the preceding Art. show that 
Eq. (4) is the only one which gives any conditions of motion in 
the problem under consideration. 



Putting the value of : 



dw 



in Eq. (4) of Art. 7, that equation may take the form : 

d'^w _ G d^iu ^ d^w jj^^"^ _ / \ 

'dF ~ m~d:!^ ' ^^' dF~ ~ dx^ ~ ^ ' ' ' ^^^ 

Q 

For brevity, b^ is written for — . 

in 

That dimension of the cross section of the body which lies 
in the direction of the radius will be assumed so small that w 
may be considered a function of x and / only. The results will 
then apply to small solid cylinders and all hollow ones with 
thin walls. 

The general Integral of Eq. (i), on the assumption just 
made, is (Booles' '' Differential Equations," Chap. XV., Ex. i) : 



Art. II.] CIRCULAR CYLINDERS. 79 

zv^-f{x-^^ht)^F{x-bt)', 

in which f and F signify a7iy ai'bitrary functiofis whatever. 
Now it is evident tliat all oscillations are of a periodic char- 
acter, />., at the end of certain equal intervals of time, w will 
have the same value. Hence since /"and i^are arbitrary forms, 
and since circular functions are periodic, there may be written: ' 

w = A^\sin (a^x -\- oijbt) -|- sin {a'^x — oiJ>t)\ 

— B^\cos {cy^x + ^i'n^t) — cos {(y^x — oiJt)i)\ ; . . (2) 

in which «'„, ^« and B^ are coefficients to be determined. 

Substituting for the sines and cosines of sums and differ- 
ences of angles : 

w = 2 sin a„x{A„ cos a^Jbt + B^ sin aj)t) ... (3) 

Let the origin of co-ordinates be taken at the fixed end of 
the piece ; w must then be equal to zero, as is shown by Eq. 
(3). But there may be other points at which w is always equal 
to zero, whatever value the time t may have. These points, 
called nodes, found by putting 2X/ = o ; or : 

sin ax = (4) 

This equation is satisfied by taking: 

_ 7t 2 TT 3 TT llTt ^ 

"m — T" > 1 ' > • • • > ■ ~ » 

a a a a 

and X ~ a\ In which a Is the length of the piece. 
Hence, at the distances : 



80 TORSIONAL OSCILLATIONS. [Art. II. 

a a a 

2 ' 3 n 

from the fixed end of the piece ^ there will exist sections which are 
7iever distorted or moved from their positions of rest. These are 
called nodes, and one is assumed at the free end, although such 
' an assumption is not necessary, since a is really the distance 
from the fixed end to the farthest 72ode and not necessarily to 
the free end. 

If, as is permissible, A^ and B^ be written for twice those 
quantities, the general value of w now becomes : 

. rrx f nbt . rrbt \ 

w = sin y4j cos 4- B, sin 

a \ a a J 

. 27lX f . 27rbt , „ . 27Tbt\ 

+ sm A 2 cos h B^ S171 ■ ) 

^ \ a ' ^ a ) 

+ sin - — A^ cos ^ + B^ sin ) 

a \ ^ a ^ a J 



, . nTTx f . nnbt , „ . nnbt\ , . 

+ sin A^ cos h Z)' sin . . (0 

a \ a a J ^^ 



The coefficients A and B are to be determined by the 
ordinary procedure for such cases, Let : 

zv^ = (p{x) 

be the expression for the initial or known strain at any point, 
for which the time t is zero. Then \( A^ is any one of the co- 
efficients A : 



Art. II.] CIRCULAR CYLINDERS. 8 1 



a 



2 C'^ , s . TtTTA: 



(p{x) sin dx (6) 



a 



The velocity at any point, or at any time, will be given by : 

dw . 7tx ( . . Ttbt „ 7tbt\ Ttb ,. 

-— — — sin — A, sin B^ cos — . . (7) 

dt a\ a a J a '^ 

In the initial condition, when the time is zero, or / = o, it 
has the given, or known, value : 



dw^ -r-, . Ttb f j^ . TIX . r^ 

— -^ = ^(x) = B. sin h 2^2 sin 

dt ^ a \ a 

+ 3^3 sm -^ + . . . y 
Then, as before : 



2nx 



/y fCt 

nnb 



^(x) sin dx (8) 



Thus the most general value of w is completely deter- 
mined. 

The intensity of shear at any place or time is given by: 

dx 

w being taken from Eq. (5). 

The second case to be treated is that of the torsion pen- 
dulum, in which the mass of the twisted body is so incon- 
siderable in comparison with that of the heavy body, or bob, 
attached to its free end that it may be neglected. 
6 



82 TORSIONAL OSCILLATIONS. [Art. II. 

Let M represent the mass of the pendulum bob, and k, its 
radius of gyration in reference to the axis about which it is to 
vibrate ; then will Mk^ be its moment of inertia about the 
same axis. 

The unbalanced moment of torsion, with the angle of 
torsion a, is, by Eq. (9) of Art. 10: 

GaTp . 

The elementary quantity of work performed by this un- 
balanced couple, if ft is the general expression for the angular 
velocity of the vibrating body, is : 

Galj, . /? dt. 

This quantity of energy is equal in amount but opposite in 
sign to the indefinitely small variation of actual energy in the 
bob ; hence : 

Gal^ft dt = - d (^^M^ = _ Mh'ft dp. 

But if a is the length of the piece twisted : 

;j = <^, and dp = "^-^^1 . 
at dt 



.-. (-^) («'^) = - Mk" 



d\cici) 
dt^ 



Multiplying this equation by 2d{aa), and for brevity put- 
ting : 



(-^)=://; {Mk^) = K', 



Art. II.] TORSION PENDULUM. 83 

then integrating and dropping the common factor a^ : 

When a = a^, the value of the angle of torsion at the ex- 
tremity of an oscillation, the bob will come to rest and — will 
be zero. Hence : 

C = Ha^, 



and 



^(i7 = ^^--'-"^)- 



•^^ ^ . dt. 



Va^' —a' \ K 

.: sm-'-=u/^ + {C' = o). ... (9) 

C = o because a and f can be put equal to zero together. 
At the opposite extremities of a complete oscillation a will 
have the values : 

(+ a,) and (- o',). 
Putting these values in the expression : 

i=^^.siu-''L (10) 

and taking the difference between the results thus obtained, 



84 THICK, HOLLOW SPHERES. [Art. 12. 

the following interval of time for a complete oscillation will be 
found : 



K Mk'a , , 



The time required for an oscillation is thus seen to vary 
directly as the square root of the moment of inertia of the bob and 
the length of the piece, and inversely as the square root of the co- 
efficient of elasticity for shearing and the polar moment of inertia 
of the normal section of the piece twisted. 

The number of complete oscillations per second is — . If 

this number is the observed quantity, the following equation 
will give G : 

^ _ , I \2 n^Mk'a 



rJ I, 



The formulas for this case should only be used when the 
mass of the cylindrical piece twisted is exceedingly small in 
comparison with M, 



Art. 12.— Thick, Hollow Spheres. 

In order to investigate the conditions of equilibrium of 
stress at any point within the material which forms a thick 
hollow sphere, it will be most convenient to use the equations 
of Art. 8. As in the case of a thick, hollow cylinder, the in- 
terior and exterior surfaces of the sphere are supposed to be 
subjected to fluid pressure. 

Let r' and rj be the interior and exterior radii, respect- 
ively. 



Art. 12.] THICK, HOLLOW SPHERES. 85 

Let — / and — /^ be the interior and exterior intensities, 
respectively. 

Since each surface is subjected to normal pressure of uni- 
form intensity no tangential internal stress can exist., but normal 
stresses in three rectangular co-ordinate directions may and do 
exist. Consequently, in the notation of Art. 8, 

J- ^r ^^ -l \lir ^^^ -l ^<f> — ^ 0« 

With a given value of r, also, a uniform state of stress will 
exist. Neither N^ nor N^ ean, then., vary with cp or ip. By the 
aid of these considerations, and after. omitting 7?^, 0^, W^, and 
the second members, the Eqs. (i), (2) and (3) of Art. 8 reduce 
to: 

dNr , 2Nr - N^ -N^ _ 

-df + ~r - o .... (I) 



- 7V^ + 7V^ = o (2) 



By Eq. (2) : 



Eq. (i) then becomes: 



dNr^ ^ Nr - N 
dr r 



-l^+,iI.^Z^=o .... (3) 



On account of the existing condition of stress, which has 
just been indicated, it at once results that : 

7] — 00 — o^ 

and that p is a function of r only. 



86 THICK, HOLLOW SPHERES. [Art. 12. 

Eqs. (4) to (10), of Art. 8, then reduce to : 



I - 2r dr ^'^ 



N,^N, =.^^d^2G^ (6) 

^ I — 2r r 



After substitution of these quantities, Eq. (3) becomes 
2Gx f d^p 2rdp— 2pdr \ , ^q^ \- aG ^^ 

- 4G^^ = o. 



or : 



dr- ^ dr 



One integration gives : 



^+^ = . = ^ (7) 

dr r 



Hence ^, the rate of variation of volume, is a constant 
quantity. Eq. (7) may take the form : 

r dp -\- 2p dr — cr dr. 



Art. 12.] THICK, HOLLOW SPHERES. 87 

As it stands, this equation is not integrable, but, by inspect- 
ing its form, it is seen that r is an integrating factor. Multi- 
plying both sides of the equation, then, by r\ 

r^ dp -[- 2rp dr = dir'^p) — cr^ dr. 

... r^f, = c~^b .-. p = ^ + ^^ . . . (8) 
3 3 ^ 

Substituting from Eqs. (7) and (8) in Eq. (5) : 

T,r '2-Gx , 2Gc AbG , . 

Nr = C -^ ^^ .... (9) 

I — 2r 3 r^ ^^^ 

^3 



It is obvious what A represents. 

When / and r^ are put for r, N^ becomes — / and — pv 
Hence: 



Ah_G 
r 



-3 ^ ' 



and : 



These equations express the conditions involved in Eqs. 
(13), (14) and (15), of Art. 6. 
The last equations give: 



^Gb = ^Ii^lllfH. 



88 THICK, HOLLOW SPHERES. [Art. 12. 

/± — r T— . 

r^ — r^ 

These quantities make it possible to express Nr and N^ in- 
dependently of the constants of integration, c and b^ for those 
intensities become : 









Thus it is seen that N^ = N^ has its greatest value for the 
interior surface ; that intensity will be called h. 

It is now required to find rj^~r' = t in terms of h, p 
and /i. 

\{r — r' in Eq. (11) : 

2/<r'3- r^) = 3Ar,3 — /(2r'3 + r/). 
Dividing this equation by r'^ and solving : 

r'3 2h- p ^ 3A * 






If the intensities / and A are given for any case, Eq. (12) 
will give such a thickness that the greatest tension Ji (suppos- 
ing /, considerably less than/) shall not exceed any assigned 



Art. 12.] THICK, HOLLOW SPHERES. 89 

value. If the external pressure is very small compared with 
the internal, p^ may be omitted. 

The values oi A and A^Gb allow the expressions for c and b 
to be at once written. 

If /i is greater than /, nothing is changed except that 
N^ — N^ becomes negative, or compression. 



CHAPTER III. 
The Energy of Elasticity. 

Art. 13. — Work Expended in Producing Strains. 

The general expressions, in rectangular co-ordinates, for the 
unbalanced forces which act in the three co-ordinate directions 
upon any indefinitely small parallelopiped of material subjected 
to any state of stress whatever, are given by multiplying each 
of Eqs. (7), (8) and (9) of Art. 6 by {dx dy dz). If an indefi- 
nitely small change in the state of stress takes place, that in- 
definitely small parallelopiped will suffer a displacement whose 
rectangular components are du^ dv, dzu ; and the amount of 
work performed in moving it will be found by multiplying each 
of the three unbalanced forces, determined as above, by each 
of the three small strains belonging to the same direction with 
the force (as in Art. 6, u, v and w are strains in the directions 
of X, y and z). This differential quantity of work, integrated 
throughout the extent of the body, will give the elementary 
quantity of work required for the small deformation and mo- 
tion of the whole body. 

The resulting equations form the foundation of investiga- 
tions in elastic vibrations and resilience ; they also furnish the 
means of reaching some general conclusions in reference to 
suddenly applied loads. 

Let <^[^F represent the elementary quantity of work required 
for the motion only, then the operations which have just been 
indicated will give the following expression : 



Art. 13-] WORK EXPENDED IN PRODUCING STRAINS. 9 1 



jr A 7- r/T /IT \ 

-~^ dx dy dz -\ — —^ dx dy dz -\ — -~ dx dy dz\du 



+ ( ~y^ dx dy dz -\- —~ dx dy dz -\ ^ dx dy dz]dv 

\dx dy dz J 



+ ( —y^ dx dy dz A 7-^ dx dy dz -\- — .-^ dx dy dz] dw 

\dx dy dz J 



+ ( ^o dii 4- Yo ^'^^ + ^o ^"^ ) dx dy dz 



m ( du —— + <^^' ^7^ + ^"^ —j^ ) dx dy dz := dW . (i) 



dl' 



de 



This equation, however, can be put in a much simpler form, 
and, caused to take a shape which will show at a glance the 
true character of each part ; dx, dy and dz are differentials of 
independent variables, hence they are arbitrary and independ- 
ent. Integrating by parts, therefore : 



[dN. 



dx 



- dx . dy dz . die = (iV/ die' — N^' du") dy dz 



yVi di -z- j dx dy dz \ 



in which the primes indicate the values of N^ and ti at one 
point of the exterior surface of the body, and the seconds those 
values for another point of the exterior surface ; these points 
being taken "at opposite extremities of a bar of the material 
whose normal section is (dy dz) and which extends entirely through 



92 



THE ENERGY OF ELASTICITY. 



[Art. 13. 



the body in the direction of x. Maintaining the same notation 
and proceeding : 



dy 



dy . dx dz . dti = ( T^ du' — T^' du") dx dz 



^3 ^ (^) dy dx dz 



dT *■ 

—j^ dz , dx dy , du = 
dz 



( T2 du' — T2' dti') dx dy 



T^di—\ dz dx dy. 



But by referring to the equations which immediately pre- 
cede (13), (14) and (15) of Art. 6, it will be seen that the sum 
of these three double integrals will represent the amount of work 
performed on tJie body by the external forces acting in the direc- 
tion of the axis of x. Precisely the same general results are 
obtained for the directions of y and z by treating in the same 
manner the remaining derivatives of the internal intensities in 
Eq. (i). The preceding operations are typical, therefore they 
need not be repeated. 

Again, by reference to the notation and demonstrations of 
Art. 5 : 

''(i)+''(S)=*^ 



■'(S+^f)-'"-.^ 



Art. 13.] WORK EXPENDED IN PRODUCING STRAINS. 



Finally 



ail -7— 



2 \dtl ' de 2 \dt J ' 



93 



^(£) = ^^' ^(S)^^^^' ^(S = ^^3 



, dHv I , fdw^ 2 
^w — r^ = — <3; ( - 



(^/^ 2 \dt J ' 



Introducing these reductions and quantities, Eq. (i) be- 
comes : 

P' da (cos n' die' -f cos x' dv' -\- cos p' dzv) 
- [P"da" (cosTt" du" + cosj:' dv" + cosp" dw") 



[ {NJi, + N,dl, + N/J^ + T^dcp^ + T,d(p, + 7;^<7?3) ^;ir dy dz 



+ 



(Xq </z^ + Y^dv -\- Zq dw) dx dy dz 



in — d 
2 






dx dy dz = dW. (2) 



Eq. (2) shows clearly the distribution of the different por- 
tions of work expended. The first two (single) integrals evi- 
dently represent the total amount of work performed by forces 
acting on the exterior surface of the body; it will be indicated 
by dW^. If the forces P' and P" are of the same kind (i.e., 
both pulls or both pushes), the algebraic sum of any two terms 



94 THE ENERGY OF ELASTICITY. [Art. 1 3. 

of these integrals will be a numerical sum if they involve co- 
sines of the same letter but of opposite signs. 

The correct application of Eq. (2) depends largely upon the 
proper observance of the signs which should affect P\ P" and 
the cosines. 

The first triple integral in the first member of Eq. (2), in 
which each intensity of stress is multiplied by the differential 
of its characteristic strain, and which will be indicated by dW^^ 
is evidently the amount of work required for the small distor- 
tion alone, of the body. The quantity within the parentheses 
is called ^ho. potential energy of the elasticity of a cubic unit of 
material^ since, if it be multiplied by (dx dy dz), the product will 
express the amount of work that small portion of material can 
perform in returning to its original condition. 

This potential energy for a cubic unit is easily integrated 
by the aid of Eqs. (11), (12), (13), (17), (18) and (19) of Art. 5. 
Making the substitutions from those equations and integrat- 
ing : 



H = 



(iV; dl, -f TV, dl, + N^ dl^ + T^dcp, -f 7; dcp^ + 7; dcp^ 

^ 3^^ ^i^ + 4^+4^ \ , ^Gr (/, + 4 + l^f 
\ 2 J \ — 2r 2 



//"is. the potential energy of a cubic unit of material for a 
change of state extending from the limit o to the strains Z^, 4, 
etc. 

The last triple integral in the first member of Eq. (2) ex- 
presses the work done by external forces which take hold of 
the mass of the body. Let it be represented by ^J^. This 



Art. 13.] WORK EXPENDED IN PRODUCING STRAINS. 



95 



triple integral added to the first two single integrals, which 
belong to the surface of the body, will give the total work 
done by external forces. 

The second member of the equation is the small variation 
of actual energy, which usually exists in consequence of vibra- 
tions. 

Let Fbe the resultant velocity of the parallelopiped, then 
will : 



1 dV = VdV =~ d 



fdu\^ fdv^ fdiv^' 
VSdtJ "^ \dtJ '^~ \dl) . 



By transferring dW^, the first two members of Eq. (2) may 
take the form : 



dW, + dW^ = dW, + mVdVdx dy dz . 



.-. W^-\-W.^=W^-\-\\\[mVdVdxdydz . . (3) 



Or, tJie total external work performed is equal to the work 
done in distorting the body added to the change of actual eji- 
ergy. 

This result expresses the law of the conservation of energy 
for the elastic bodies considered. 

If the external work is nothing, the first member of Eq. (3) 
is zero. The actual energy will then exist in consequence of a 
state of vibration. Let its variable value be represented by U, 
Since dx., dy, and dz are arbitrary : 



U=^ 



ni — dx dy dz — C \ 
2 



9^ THE ENERGY OF ELASTICITY. [Art. 1 4. 

C representing a constant of integration. Under the circum- 
stances assumed, then : 

W,+ U=C (4) 

Hence, the total e^iergy of the vibrating body (i.e., the sum of the 
actual and potential^ will be constant. 



Art. 14. — Resilience. 

The term resilience is appHed to the quantity of work which 
is required to be expended in order to produce a given state of 
strain in a body. The analytical expression for this amount of 
work is obtained directly from Eq. (2) of the preceding Art. 

Let the simple case of a single straight bar be considered ; 
and let all the external forces act parallel to the axis of the bar 
while they take hold of the end surfaces, which are normal 
sections. These external forces will be considered equal to the 
internal stresses developed ; consequently no vibrations will 
exist. The action of the external forces X^, Y^ and Z^ will also 
be omitted. 

Now, if the axis of x be taken parallel to the axis of the 
bar, and if that end of the bar to which P" is applied be fixed, 
there will result from the preceding conditions : 

cos 7t' = cos 7t" ^= I, 

cos X = cos p' — cos x" = cos p" = du" = O, 

N, = N^= T,= T,= T^ =0. 

Eq. (2) will then become : 



P'da du' = 



TVj dl^dx dy dz . . . . (l) 



Art. 14.] RESILIENCE. 97 

But if the intensity P' is uniform and A the area of normal 
section, Eq. (i) becomes : 

P'A du' = AN, x,dl, (2) 

in which x, is the length of the bar. 
From Eq. (i) of Art i : 

hence : 

f r^i /= 

FA du' — Resilience = Ax, E /, dl, = Ax,E^ . (3) 

J Jo 2 

I 

The quantity : 

El^ — -— 

is called the ^^ Modulus of Resilience.'' This term is usually 
applied when N, is the greatest intensity allowed in the bar. 

If one end of a bar, placed in a vertical position, is fixed, 
while a falling body whose weight is iv^ acts upon the other 
end, the height of fall may be sufficient to produce rupture. 
Let h be the height of fall required and N,—p the ultimate 
resistance of the material of the bar. In order that rupture 
may take place : 



S2 



, Ax, f 1 A f . . 

wh = — - . ^ ,'. /i = X, — ' "j^ ' . . . (4) 

2 E 2w E ^ 



Eq. (4) shows that the height of fall varies directly as the 
lerigth of the piece. It is virtually assumed, however, that the 
extension or compression is uniform throughout the length of 

7 



98 THE ENERGY OF ELASTICITY. [Art. 1 5. 

the bar, to the instant of rupture. This, in reab'ty, is not true, 
and h will not vary as rapidly as x\. The principle established 
in Eq. (4) is equally true for torsion and bending. 



Art. 15. — Suddenly Applied External Forces or Loads. 

A very important deduction can be reached by an attentive 
consideration of Eq. (2) of Art. 13, if it be assumed that the 
external forces P' and P'' are simple and direct functions of 
the external strains ii, v and w. In such a case the following 
relations will hold, in which ^, b and c are constants : 

P' cos n' = au ; P' cos j(' z=z bv' \ P' cos p' = cw' ; 
P" cos tt" = au' ; P" cos x" = bv" ; P" cos p" = cw" . 

Consequently the external work performed, om.itting X^, Y^ 
and Z^, in changing the body from a state of no stress to that 
indicated by the strains n', t)', to', n" ^ t)"^ tn", will be : 



dW, 



da' (a \- b — -\~ c — ■) 

\ 2 2 2 / 



da"[a'^-^b'^-^c^)=^ W; 



in which equations the integrals are to be made to cover the 
whole extent of the surface. 

If, instead of being variable, the forces P' and P" are con- 
stant and equal to the ^nal values of the preceding case (?>., 
equal to an', bv', cm', an", etc.), the external work performed in 
bringing the body to the final state u', t)', etc., will be : 



Art. 15.] 



SUDDENLY APPLIED LOADS. 



99 



dW^ = 



da («u'^ -t- ^0'^ + ^to'O 






2W\ 



This last case Is that of '' suddenly " applied external forces 
or loads, while the former is that of gradual application, in 
which the external forces, at each instant, are equal to the in- 
ternal resistances. In the case of sudden application it is seen 
that the amount of work expended is twice as great as in the 
other case ; consequently when the body arrives at the state of 
strain indicated by ii', t)', etc., there remains to be expended just 
as mucJi zuork as has already been performed, and at the instant 
in question it exists in the body in the shape of actual energy. 

But if an amount of energy equal to W will produce the 
strains u', t)', etc., and if, while the force acts which performed 
the work, an additional amount of energy equal to W be ex- 
pended on the body, additional strains equal to u', t)', etc., will 
be produced in the body. 

When the body comes to rest, therefore, the external strains 
will be 2Xt', 2tj', 2tD', etc. There is then no actual energy, all is 
potential. 

Since the external strains are 2u', 2tJ', etc., the external 
work which has been performed up to this instant will be found 
by putting those quantities in the place of tt', d', etc., in the 
expression for W\ above. That expression will then become 

For gradually applied loads Eq. (2) of Art. 13 becomes 
simply : 



W = 



H dx dy dz ; 



in which //"is the potential energy per cubic unit for the state 



100 THE ENERGY OF ELASTICITY. [Art. 1 6 

of strain corresponding to u', t)', to', etc. But, if the loads be 
suddenly applied, in accordance with what has been given, the 
Eq. (2) of Art. 13 becomes : 

4 W — \H dx dy dz , 

Now the expression for H, given in Art. 13, shows that 
multiplying H hy 4. is the saute thing as doubling the strains : 

h^ 4, 4' *5^i' ^2 ^"d ^3- 

But by doubling the strains the intensities of stresses are 
doubled. Hence, if the same loads are first applied gradually 
and then suddenly, the strains and stresses in the latter case will 
be double those in the former. This is a very important principle 
in engineering practice, for it covers all cases of tension, com- 
pression, torsion and bending. It also finds many important 
extensions in special cases of such structures as iron and steel 
bridges, particularly suspension bridges. For the considera- 
tions involved in this Art. show that in all cases of sudden 
applications of loads, actual energy will be stored and restored 
during different intervals of time and, consequently, that vibra- 
tions will be initiated. 

Eq. (2) of Art. 13 furnishes a most convenient and elegant 
point of departure for investigations in such special cases, as 
will be exemplified in the next Art. 



Art. 16, — Longitudinal Oscillations of a Straight Bar of Uniform 

Section. 

The complete solution of this problem will not be given, 
though it may be reached. 

Let the bar be fixed at one end in a vertical position and 



Art. 1 6.] 



LONGITUDINAL OSCILLA TIONS. 



lOI 



let a heavy weight, Wy act on the other. Also, let the axis of 
X be taken parallel to the axis of the bar, whose uniform nor- 
mal section will be represented by A. 

On account of the circumstances of application of the ex- 
ternal forces and position of bar, the following equations of 
condition will exist : 

cos x' = <^os p' = cos x" = ^os p" = die" = N'^ = ^3 
= T, = T^ = T^ = o = Y^ = Z^, 



dv 
di 



dw 
~di 



du 
will be very small compared with — , hence they will be 

omitted. P' is the heavy weight attached to the free end of 
the bar divided by A ; consequently : 



cos 7t 



I. 



Eq. (2) of Art. 13, now reduces to : 



P'dd dlL - 



El^ dl^ dx dy dz -\- X^ du dx dy dz 



in — d [ —r) dx dy dz 

2 \dt I -^ 



(0 



The integrals are to be extended throughout the whole of 
the bar. Since strains and stresses are uniform for any one 
cross section of the bar, and because X^= w — weight of a 
unit of volume of the bar (the force of gravity is the only ex- 
ternal force which acts on the mass of the bar), Eq. i be- 
comes : 



I02 THE ENERGY OF ELASTICITY. [Art. 1 6. 

Wdv! - AE^^dx + Awu dx = - A (^ dx -\- C dx . (2) 

This equation {C being a constant of integration) involves 
the complete problem of longitudinal oscillations. Two spe- 
cial cases, only, however, will be treated, in which the weight of 
the bar is so small compared with ^that it may be neglected. 
This condition involves the omission of : 

iH / dti\ ^ 
Awu dx and — A(-z-] dx y 
2 \dt ) ' 

in Eq. (2), and makes l^ constant throughout the length of the 
bar. 

Since the equation must be homogeneous, C will represent 
a quantity of actual energy ; in fact, a part of tJiat quantity 
stored^ at any instant^ in W. 

If x^ represents the length of the bar, C may be put equal 
to : 

W fdu\ 
2gx, \dt J ' 

Also, because /j is constant for the whole bar : 



, _ 2/ 

Introducing all these changes in Eq. (2) and integrating : 

Wu'-AE'^ = ^(^y-^C'. ... (3) 

2X, 2g \dt) ^^^ 

If W is suddenly applied to the bar while in a state of 
equilibrium or rest, for which : 



Art. 1 6.] LONGITUDINAL OSCILLATIONS. IO3 

, dll 

C will be zero, as the equation shows by such a substitution. 
For this case Eq. (3) becomes, after omitting the primes : 



dt — 



_ / Wx\ die 



A Eg /2Wx 



EA 



- u' — te 



/ ^y^\ ■ _i AEu , . 



The limits of the amplitude are discovered by putting 



-7- (the velocity) = o, 



in Eq. (3), remembering that C is also equal to zero. That 
operation will give : 

, 2Wx, 

u = O and u = — rr^ * 

AE 

Putting these values in Eq. (4), successively, and taking the 
difference of the results, the time occupied by one oscillation 
will be : 



^^•^/Z^ "'''' ^"'"" ^ = '"VJ • • ^5) 



in which equation : 



I04 THE ENERGY OF ELASTICITY. [Art. 1 6. 

is the strain in the bar caused by a gradual application of W. 

In the second case to be treated the bar is first supposed to 
take a vertical position, with the weight attached to its free 
end, in a state of equilibrium. An external force then de- 
presses the free end a distance zi^, measured from its position of 
equilibrium. If the force F is now removed, the weight will 
make excursions on each side of its position of rest. 

Let 2/j represent the value of u! corresponding to the weight 
W alone, as in the previous case ; then let : 

U! = Uj. -\- Uy 

u being measured from the position of equilibrium of the 
weight W. 

Eq. (3) will then take the form : 

T(rf , \ ^E, / , N, H^ fdiu, + liSV , ^ /^x 
W{u, + «) - — («. + uj = - (^i^7 + ^- (6) 

When u = u^ the body comes to rest. Hence : 

W{u, + «„) - ^ («, + u^' =C. . . . (7) 

Subtracting Eq. (7) from Eq. (6) : 

iirr \ AE r / x , o-. ^^ /dziY /ox 

^'' - ''o) - — [2?A (^^ - ?0 + ^' - ^^0] = - [-^J • (8) 

since : 

d(Uj^ -\- u) = du. 
Remembering that : 



Art. 1 6.] LONGITUDINAL OSCILLATIONS. IO5 

A Ell 
W{u - u^ = — ^-i {u - ?/„), 



Eq. (8) may take the form : 



7. / ^^'x du 

at = ' 



A Eg ^u^ _ ^^2 



(9) 



••• ^ = A/~-^^*^^"'!r (10) 



<r ^^c 



Eq. (9) shows that the ampHtude of a vibration is found by 
putting : 

?/ = + Uq or — ?/o. 

Putting these values in Eq. (10) and taking the difference 
of the results, the time of a single oscillation is found to be : 



T=^xr-^ (II) 



Eq. (11) is seen to be identical with Eq. (5). In this case 
the amplitude is 211^, and the body oscillates through its posi- 
tion of rest. Both oscillations are completely isochronous for 
the same weight W. 

If n is the observed number of oscillations per second, 
either Eq. (5) or (i i) gives : 

E — ^=r . — 1, — - — n^ — 1, — -■ ' . . . (12) 
T- Ag Ag ^ ^ 

from which E may be computed, if W is very great compared 
with the weight of the bar or wire. 



CHAPTER IV. 

Theory of Flexure. 

Art. 17. — General Formulae. 

If a prismatic portion of material is either supported at both 
ends, or fixed at one or both ends, and subjected to the action 
of external forces whose directions are normal to, and cut, the 
axis of the prismatic piece, that piece is said to be subjected to 
*' flexure." If these external forces have lines of action which 
are oblique to the axis of the piece, it is subjected to combined 
flexure and direct stress. 

Again, if the piece of material is acted upon by a couple 
having the same axis with itself, it will be subjected to " tor- 
sion." 

The most general case possible is that which combines these 
three, and some general equations relating to it will first be 
established. 

The co-ordinate axis of Jf will be taken to coincide with the 
axis of the prism, and it zvill be assumed that all external forces 
act upon its ends only. Since no external forces act upon its 
lateral surface, there will be taken : 

retaining tne notation of Art. 6. These conditions are not 
strictly true for the general case, but the errors are, at most, 
excessively small for the cases of direct stress or flexure, or 



Art. 17.] GENERAL FORMULAE, 10/ 

for a combination of the two. By the use of Eqs. (12), (21) 
and (22) of Art. 5, the conditions just given become : 

r fdti j^ dv dw\ . dv __ , . 

I - 2r \dl' dy'^ d^J ^ dy ~^ ' ' ' ^^^ 



r fdu dv dw\ dw _ . 

I - 2r \dx'^ ly'^ ~dz) ^ 'd^~^ ' ' • ^^^ 



dv , dzt> , . 



Eqs. (i) and (2) then give : 

dv dw , . 

dy-d^=° (4) 

In consequence of Eq. (4), Eqs. (i) and (2) give : 



dv __ dw _ du , . 

dy~dz dx '''■''' ^ 



By the aid of Eq. (5) and the use of Eqs. (11), (13) and (20) 
of Art. 5, in Eqs. (10), (11) and (12) of Art. 6 (in this case 
Xq = Vq = Zq =z o), there will result : 



d^u , d^?i , d^u ,^v 



d^ii d^v _ . . 

d^ + 5^ - ^ • • ° ' • • • ^^^ 



I08 THEORY OF FLEXURE. [Art. 1 7. 



dx dz dx' 

The Eqs. (3), (5), (6), (7) and (8) are five equations of con- 
dition by which the strains u, v and w are to be determined. 
Let Eq. (6) be differentiated in respect to x : 

d^u d^u d^u _ 

dx"^ dy^ dx dz^ dx 

From this equation let there be subtracted the sum of the 
results obtained by differentiating Eq. (7) in respect tojj^ and 
(8) in respect to ^ : 

dHi d^v dHv _ 

dx'^ dx^ dy dx^ dz 



In this equation substitute the results obtained by differ- 
entiating Eq. (5) twice in respect to x^ there will result : 



^3 dif) 

dht \dxj , . 

^3 = -^nr = ° (9) 



This result, in the equation immediately preceding Eq. (9) 
by the aid of Eq. (5), will give : 

d^v 

- = o. 



dx'^ dy 



After differentiating Eq. (7) in respect to j/, and substi- 
tuting the value immediately above : 



Art. 17.] GENERAL FORMULA. IO9 



d^u \axj 



dy^ dx dy^ 



o (ic) 



Eqs. (9) and (10) enable the second equation preceding Eq. 
(9), to give : 

df—\ 
d^u \dx) r . 

— ^ ^ = o (11) 



ds^ dx dz' 

Let the results obtained by differentiating Eq. (7) in re- 
spect to z and (8) in respect to j, be added : 

d'^ii , d^v , d^iv 

+ ~i r + -7 r- = o. 



dx dy dz dx^ dz dx^ dy 

The sum of the second and third terms of the first member 
of this equation is zero, as is shown by twice differentiating Eq. 
(3) in respect to x. Hence : 

dhi \dxj , . 

= O . . . . . (12) 



dy dz dx dy dz 



The Eqs. (9), (10), (11) and (12) are sufficient for the de- 
mination of the f( 
to be algebraic, for : 



dti 
termination of the form of the function -7-, if it be assumed 

dx 



Eq. 


(9) 


shows 


that x^ does 


not appear in it ; 


<< 


(10) 


u 


'■' f " 


ii a <( 


H 


(") 


<( 


(( ^2 (I 


<< it (( 


H 


(12) 


cc 


" yz '' 


a (( u 



no THEORY OF FLEXURE. [Art. I/. 

The products xz and xy may, however, be found in the 
function. Hence if a^ a^j a^j b, b^^, and b^ are constants, there 
may be written : 

dtc 

—_ = a -\- a^z -\- a^ -\- X {b Ar b^z -Y b^y) . . (13) 

Eq. (5) then gives : 

dv dw f , , / r , , r \ 5 / \ 

dy"^ ~dz^ ~ '^ ^^ '^ ^^^ '^ ^^^ ~^ ^ ^ '^ '^ ~^ ^-^'^^ ^^"^^ 

Substituting from Eq. (13) in Eqs. (7) and (8) : 

d^v 



dx' 



= - a,- b,x (15) 



d^w 

— — — — a. — b,x (16) 

dx^ ^ ^ ^ ^ 



The method of treatment of the various partial derivatives 
in the search for Eqs. (13) and (14) is identical with that given 
by Clebsch in his *' Theorie der Elasticitdt Fester Kdrper'' 

It is to be noticed that the preceding treatment has been 
entirely independent of ih.Q form of cross section or direction of 
external forces. 

It is evident from- Eqs. (13) and (14), that the constant a 
depends upon that component of the external force which acts 
parallel to the axis of the piece and produces tension or com- 
pression only. For, by Arts. 2 and 3, it is known that if a 
piece of material be subjected to direct stress only : 

du . dv dw 

-— = a and —- = ~~ = — ra; 

ax dy dz 



Art. 17.] GENERAL FORMULA. Ill 

the negative sign showing that ra is opposite in kind to a, both 
being constant. 

Again, if z and y are each equal to zero, Eq. (13) shows 
that : 

du , , 

-y- ^=^ a -\- OX , 
ax 



Hence bx is a part of the rate of strain in the direction of x 
which is uniform over the whole of any normal section of the 
piece of material, and it varies directly with x. But such a 
portion of the rate of strain can only be produced by external 
force, acting parallel to the axis of X, and whose intensity 
varies directly as x. But, in the present case such a force does 
not exist. Hence b must equal zero. 

The Eqs. (13), (14), (15) and (16), show that a^, b^^ and a^, b^ 
are symmetrical, so to speak, in reference to the co-ordinates z 
and y, while Eqs. (13) and (14) show that the normal intensity 
N^ is dependent on those, and no other, constants in pure 
flexure, in which a ^= o. It follows, therefore, that those two 
pairs of constants belong to the two cases of flexure about the 
two axes of Z and Y. 

No direct stress N^ can exist in torsion, which is simply a 
twisting or turning about the axis of X. 

Since the generality of the deductions will be in no manner 
affected, pure flexure about the axis of Fwill be considered. 
For this case : 

a = a^ =^ b^ = o — b. 

Making these changes in (13) and (14) : 

du , J , . 

—^^a.z^b^xz (17) 



112 THEORY OF FLEXURE. [Art, I/. 



dv dw du f I 7 \ /.o\ 

.-. e = $'+$+ ^ = . (.. + K.) (I - 2.) . (19) 



Also : 



dx dy dz 



,^ 2Gr ^ , ^<^2/ 



I — 2r ^^' 

.-. TV, = 2 6^(r + i) (^, 4- ^,;i> = E{a, + <5,^> . (20) 

since : 

2G{r + i) = E. 

Taking the first derivative of jV, : 
dJV. 



dz 



= ^(^, + h^x) (21) 



This important equation gives the law of variation of the 
intensity of stress acting parallel to the axis of a bent beam, in 
the case of pure flexure produced by forces exerted at its ex- 
tremity. That equation proves, that in a given normal section 
of the beam, whatever may be the form of the section, the rate 
of variation of the normal intensity of stress is constant ; the rate 
being taken along the direction of tJie external forces. 

It follows from this, that N^ must vary directly as the dis- 
tance from some particular line in the normal section consid- 
ered in which its value is zero. Since the external forces F are 
normal to the axis of the beam and direction of N^^ and be- 
cause it is necessary for equilibrium that the sum of all the 
forces N^ dy dz, for a given section, must be equal to zero, it 



Art. 17.] 



GENERAL FORMULA. 



113 



follows that on one side of this line tension must exist and on 
the other, compression. 

Let N represent the normal intensity of stress at the dis- 




Fig.l 




tance unity from the line, b the variable width of the section 
parallel to j, and let A = b dz. The sum of all the tensile stress 
in the section will be : 



NzA ~ N\ zi\ 



The total compressive stress will be 



N 



A 



The integrals are taken between the limits o and the greatest 
value of z in each direction, so as to extend over the entire 
section. In order that equilibrium may exist therefore : 



N 



Co 



•A 4- 



A y = o. 



ZA=0 
"i 



(22) 



114 



THEORY OF FLEXURE. 



[Art. 17. 



Eq. (22) shows that the line of no stress must pass through the 
centre of gravity of the normal section. 

This line of no stress is called the neutral axis of the section. 
Regarding the whole beam, there will be a surface which will 
contain all the neutral axes of the different sections, and it is 
called the neutral surface of the bent beam. The neutral axis 
of any section, therefore, is the line of intersection of the plane 
of section and neutral surface. 

Hereafter the axis of X will be so taken as to traverse the 
centres of gravity of the different normal sections before flex- 
ure. The origin of co-ordinates will then be taken at the 
centre of gravity of the fixed end of the beam, as shown in 
Fig. I. 

The value of the expression {a^ -\- b^x), in terms of the ex- 
ternal bending moment, is yet to be determined. Consider 
any normal section of the beam located at the distance x from 
O, Fig. I, and let OA = I. Also let Fig. 2 rep- 
resent the section considered, in which BC is 
the neutral axis and d' and ^^ the distances of 
the most remote fibres from BC. Let moments 
of all the forces acting upon the portion {l—x) 
of the beam be taken about the neutral axis 
BC. If, again, d is the variable width of the 
Fig.2 beam, the internal resisting moment will be : 




'd' 



iVj bz dz —E(a^ -f ^i-'^) 



z^ . b dz. 



— d. 



But the integral expression in this equation is the moment 
of ijiertia of the normal section about the neutral axis^ which 
will hereafter be represented by /. The moment of the 
external force, or forces, /% will be F{l—x) and it will be 
equal, but opposite in sign, to the internal resisting moment. 
Hence : 



Art. 17.] GENERAL FORMULy^. ' ' 1 15 

F{l-x) = Af = - E {a, + b,x)I . . . (23) 

M 

.-. - {a, J^ b,x) = ^j- . • . . . . (24) 



Substituting this quantity in Eq. (16) : 

d^w M 



dx^ EI 



(25) 



It will hereafter be seen that Eq. (25) is one of the most 
important equations in the whole subject of the ''^Resistance of 
Materials y 

An equation exactly similar to (25) may, of course, be 
written from Eq. (16) ; but in such an expression J/ will repre- 
sent the external bending moment about an axis parallel to 
the axis of Z, 

No attempt has hitherto been made to determine the com- 
plete values of u^ v^ and w^ for the mathematical operations 
involved are very extended. If, however, a beam be considered 
whose width, parallel to the axis of F, is indefinitely small u 
and w may be determined without difficulty. The conclusions 
reached in this manner will be applicable to any long rectan- 
gular beam without essential error. 

If y is indefinitely small all terms involving it as a factor 
will disappear in it and w \ or, the expressio7is for the strains u 
and IV will be functions of z ajid X only. But making u and ze^ 
functions of z and x only is equivalent to a restriction of lateral 
strains to the direction of z only, or, to the reduction of the 
direct strains one half, since direct strains and lateral strains in 
two directions accompany each other in the unrestricted case. 
Now as the lateral strain in one direction is supposed to retain 
the same amount as before, while the direct strain is considered 
only half as great, the value of their ratio for the present case 



Il6 ■ THEORY OF FLEXURE. [Art. 1 7. 

will be twice as great as that used in Arts. 3 and 4. Hence 2r 
must be written for r, in order that that letter may represent 
the ratio for the unrestricted case, and this will be done in the 
following equations. 

Since w and u are independent of/ : 

dw du . ^ ^dv 

-j— = — - = o, and -L ■.— ^ -r * 
ay ay ^ ax 

But by Eq. (14) : 

V — — 2r (a^ ^ b^x)zy + f(x, £), 

By Eq. (3), since : 

dw __ 
dy 

— = - 2r{a^-\- b,x)y + ^/(^', s) = O. 

This equation, however, involves a contradiction, for it 
makes /(^', z) equal to a function which involves j/, which is 
impossible. Hence : 

/(x, £) = o. 
Consequently : 

— = - 2r{a,-\- b^x)y ; 
which is indefinitely small compared with : 

— r= — 2r («j + b^x)z, 

and is to be considered zero. 



Art. 17.] GENERAL FORMULA. 1 17 



Because y(;ir, ^s") = o : 



dv - 

ax 



This quantity is indefinitely small ; hence : 

T^—— 2Grb^zy 

is of the same magnitude. 

Under the assumption made in reference to j, there may 
be written from Eqs. (17) and (18) : 

u = a^xz -t b^ — z -\- f'{z) (26) 

w = — r{a^z^ 4- b^xz^) + f{x) .... (27) 
Using Eq. (26) in connection with Eq. (6) : 

By two integrations : 

/'(^) - - ^ - ^'." + ^" (28) 

Using Eq. (27) in connection with Eq. (8) : 



dx^ 



dV{x) _ _ 
dx- " ' 

By two integrations : 



Il8 THEORY OF FLEXURE, [Art. 1 7. 

f{x) = - b,~ ^- + ^,;r + <-„. 

The functions u and w now become : 

x^ b z^ 

u = a^xz -\- b^ — z ? — —c'z -\- c" , . . (29) 

2 3 

^1'^ {Z x^ 

w = — ra^z^ — rb^xz^ "~ ^^ V "" ~ 1~ ^^^ ~J~ ^" • (3^) 



The constants of integration c\ c'\ etc., depend upon the 
values of u and w^ and their derivatives, for certain reference 
values of the co-ordinates x and z, and, also, upon the manner 
of application of the external forces, Fy at the end of the beam. 
Fig. I. The last condition is involved in the application of 
Eqs. (13), (14) and (15) of Art. 6. 

In Fig. I let the beam be fixed at 0. There will then re- 
sult for X ~o and z — q-. 



du _ 
dz 



x-=o 



ill = o, and w = 6) 

In virtue of the last condition : 

c" = c,, = o. 
In consequence of the first : 

c' = o. 
After inserting these values in Eqs. (29) and (30) 



Art. I/.] GENERAL FORMULA. IIQ 

dzv x^ 

The surface of the end of the beam, on which F \?> applied, 
is at the distance / from the origin O and parallel to the plane 
ZY. Also the force F has a direction parallel to the axis of 
Z. Using the notation of Eqs. (13), (14) and (15) of Art. 6, 
these conditions give : 

cos p = I, cos g = o, cos r = Oy 

cos TT = Oy COS X = Oy COS p = I, 

Since for x = I : 

M = F{1 - x) = o, 

Eqs. (24) and (20) give N^ = o for all points of the end sur- 
face. Eq. (15) is, then, the only one of those equations which 
is available for the determination of c^. 
That equation becomes simply : 

7; = P. 

For a given value of ^, therefore, any value maybe assumed 
for T^. For the upper and lower surfaces of the beam let the 
intensity of shear be zero ; or for 2 = ± d let T2 = o. Hence, 
by Eq. (31) : 



120 THEORY OF FLEXURE. [Art. I/. 
.-. 7; ^^±{d^- Z-) (32) 



The constants a^ and b^ still remain to be found. The only- 
forces acting upon the portion (/ — x) of the beam, are i^and 
the sum of all the shears T^ which act in the section x. Let 
Aj/ be the indefinitely small width of the beam, which, since 
z is finite, is thus really made constant. The principles of 
equilibrium require that : 



T2 . AjK . dz = Gb^ (i + r) (d^ . Ajk , dz — z^ . i\y . dz)^= F, 



+ d 

J 



The first part of the integral will be 2Aj/d^ and the second 
part will be the moment of inertia of the cross section (made 
rectangular by taking A/ constant) about the neutral axis. 
Hence : 



(33) 
(34) 



f^ • Of /' — 


F 


F 




2G{i + r)/ ~ 


EI 


••• T. = ^^(d'- 


-.-).... 


. 



\{ X — o in Eq. (24) : 



Fl . , 

^r= -;^ (35) 



Thus the two conditions of equilibrium are involved in the 
determination of a^ and b^. The complete values of the strains 
u and w are, finally : 



Art. 17.] GENERAL FORMULA. 121 

F f x^ lx^\ Fd^x 

^^ = -^-{^lr.--r.^-- + ^)+-^ . . (37) 

These results are strictly true for rectangular beams of in- 
definitely small width, but they may be applied to any rectan- 
gular beam fixed at one end and loaded at the other, with 
sufficient accuracy for the ordinary purposes of the civil en- 
gineer. It is to be remembered that the load at the end is 
supposed to be applied according to the law given by Eq. (34) ; 
a condition which is never realized. Hence these formulae are 
better applicable to long than short beams. 

The greatest value of T^, in Eq. (34), is found at the neutral 
axis by making z = 0] for which it becomes : 

7; = i^ = l.^ (38) 

2I 2 2d ^^ -^ 

F , 

— % is the mean intensity of shear in the cross section ; hence, 

the greatest intensity of shear is once and a half as great as the 
mean. 

In Eq. (36) if ^ == o, u — o. Hence no point of the neutral 
surface suffers longitudinal displacement. 

In Eq. (37) the last term of the second member is that part 
of the vertical deflection due to the shear at the neutral sur- 
face, as is shown by Eq. (38). The first term of the second 
member, being independent of -r, is that part of the deflection 
which arises wholly from the deformation of the normal cross 
section. 

The usual modification of the preceding treatment, designed 
to supply formulae for the ordinary experience of the engineer, 
will be given in the succeeding Arts. 



122 THEORY OF, FLEXURE. [Art. 1 8. 



Art. i8. — The Common Theory oi Flexure. 

The *' common theory " of flexure is completely expressed 
by Eq. (25) of Art. 17. That equation involves the condition 
that no external force acts upon the exterior surface of the bar or 
beam. In reality this condition is never fulfilled. External 
loads are applied in any manner whatever, causing normal 
compressive stresses to exist at any or all points of the ex- 
terior surface. It is assumed in the common theory of flexure 
that the equation : 

d^ ~ EI ^^^ 



holds true, for pure bending, whatever may be the number or 
manner of application of the external forces or loads. 

By " pure bending " is meant the action of external forces 
whose directions are normal to, and cut, the axis of the beam. 

As has already been seen in Art. 17, w, strictly speaking, is 
a function of x, y and z. 

It is further assumed in the common theory of flexure that w 
is a function of x only. 

This is equivalent to an assumption that the lateral dimen- 
sions of the piece are so small that they can have no influence 
on the value of w, and consequently that they will not appear 
in it. In other words the common theory of flexure is the 
theory of the flexure of pieces, one or two of whose cross 
dimensions are indefinitely small in comparison with their length. 
The neglect of this fact has led to some erroneous applications 
and deductions in connection with long column formulae. 

Eq. (i), taken in connection with these two important as- 
sumptions, constitutes the "'* Common Theory of Flexure," 
which is always used in engineering practice. 



Art. 1 8.] THE COMMON THEORY OF FLEXURE. 1 23 

Since the intensity of external loading is almost invariably 
very small compared with the internal stress N^, the first of the 
above assumptions involves very little error in all ordinary 
cases. 

The second assumption, as was stated above, is equivalent 
to taking the bar or beam so small that the strain or "■ deflec- 
tion " w is essentially the same at all points of a given cross 
section. With such small strains and large ratios of length to 
lateral dimensions as almost always occur, this assumption, 
also, involves no considerable error. 

It is well known that if the curvature is very small, the re- 
ciprocal of the radius of curvature, in the plane zx., is repre- 

sented with no essential error by — — . Hence Eq. (i) may 



dx^ 



take the form : 

EI 



= M . (2) 

P 



in which p is the radius of curvature. 

Let M' and Mj represent two bending moments which will 
produce the two radii of curvature p' and p^, Eq. (2) will then 
give the following : 

v = ^' . • • (3) 



EI 

^ = ^. (4) 



Hence : 



^\j:- 7^ = ""''-''' (5) 



124 



THEORY OF FLEXURE. 



[Art, 1 8. 



The second member shows that a bending moment : 

M, - M' r= M, ■ 

applied to a curved beam whose radius of curvature at any 
section is p', will produce a change of curvature expressed by : 



Pi 



P 



In other words : f/ie common theory of ficxttre is applicable to 
curved beams of slight curvature. 

In such a case — , Eq. (2), expresses the variation (increase 

or decrease) of curvature caused by the mom.ent M. It is to 
be distinctly borne in mind, however, that Eq. (2) itself is 
made approximately true only by considering the curvature 
very small. 

The limits within which the common theory is applicable to 

curved beams, and the degree 
of approximation of the appli- 
cation, will be shown by the 
following investigations, in 
which the longitudinal com- 
pression and extension, due to 
the external forces, w^ill be 
neglected. 

In the figure let a portion 
of any curved beam, whose 
lateral dimensions are small 
compared with its length, be 




Fig.1 



represented. Let AB represent an indefinitely short length, 
ds, of the neutral surface. C is the centre of curvature of ds 
before flexure, and C the same point after flexure. Since the 



Art. 1 8.] THE COMMON THEORY OF FLEXURE. 1 25 

lateral dimensions are small compared with the length, if the 
strains are not great, any normal cross section may, without 
essential error, be taken as plane after flexure, and such planes 
passing through A and B will then contain the radii of curva- 
ture at the points A and B. Let : 

AC = p' and AC = p 
also : 

Aa = Ab = Be — Bd = unity. 

« 

Aa and Bd are the positions taken by Ad and Be after flexure. 

The angle, before flexure, between two radii A C and BC, in- 

ds 
definitely near to each other, is — ; after flexure, as the figure 

shows, the same angle becomes — y. Hence the change in 

curvature (or change of angle between consecutive radii) 
caused by flexure is : 

ds i-j- 

\p P 

Now let the amount of shortening or lengthening of a unit 
of length of fibres, parallel to the neutral surface and situated 
at unit's distance from it, be represented by w, concisely stated, 
u is the rate of strain for any point at unit's distance from the 
neutral surface. In the figure, the amount of strain for 
AB = ds is: 

ad -\- ed = u ds. 

But the difference between the angles aC'd and bCe is : 
{ab -\- ed) -- Ab = ab -\- ed = u ds. 



126 THEORY OF FLEXURE. [Art. 1 8. 

But this difference is the change of curvature ; hence : 

^ = -1 - -i ....... (6) 

This relation is purely kinematical ; a value for u must next 
be determined in terms of the bending moment M. 

Under the circumstances of the case it has been seen that 
the longitudinal strains parallel to the neutral surface vary es- 
sentially directly as their distances from it (this law is the as- 
sumption that plane normal sections before flexure are also 
plane afterwards). The strain at any distance z from the 
neutral surface will then be uz. But it was shown in Art. i/ 
that the intensity of longitudinal stress iVj varies directly as z ; 
hence there may be written : 

A\ = Euz. 

If b is the variable width of cross section, taken parallel to 
the neutral surface, the internal resisting moment of the sec- 
tion will be : 



M = 



N^ b dz , z = Eu 



bz^ dz. 



.-. M = Eul (;) 

'^ ""-EI (^) 

The integration is to be extended over the whole section. 
Then, if the "• neutral axis " is the line of intersection of the 
neutral surface with the normal section^ I is the moment of 
inertia of the normal section about the neutral axis. 



Art. 1 8.] THE COMMON THEORY OF FLEXURE, 12/ 

Eqs. (6) and (8) then give : 

M I I , . 

-Erj-~p (9) 

This equation is true, under the assumption made, for any 
degree of curvature whatever in the original beam. 

If w and X are rectangular co-ordinates in the plane of the 
beam, x being the independent variable, the expressions for 
the reciprocals of the radius of curvature before and after 
flexure, are : 



(10) 

(II) 

By the binomial formula : 

/ dw^\ v_ 3 dw^ 15 ^7e',4 

K + ~d^) ~ ' ~ 2 ^:^^ + T ^x^~ - ^^"•' • 

and an exactly similar expression for —7. After introducing 

these in Eqs. (10) and (11), and supposing the deflections to be 
small, there may be written : 

I I _ dhu' d'w, 3 fdzv^^ dw'^\ d^w' 



I dhi', ( dw^\ 2 

p - dx^ K "^ dx^) * • 


• • • 


I d^'w' / dw''X'\ 


• • • 



P' 9 dx"" dx^ 2 \dx^ dx^J dx 



¥ \dx^ ~ ~d^) 'dx^ " ^^^* 



128 THEORY OF FLEXURE. [Art. 1 8. 

If, in addition to small deflections, the values of : 

dw' . dzt\ 

dx^ dx^ 



are not great, the equation just written shows that with a con- 
siderable degree of approxiriiation : 

I _ I _ d^'w' _ d^'w^ 
^ ~ 7" ~ ~d^ ~ ~~d^ ^^^ 

The smaller the curvature the more nearly accurate is Eq. 
(12). If, as before, w is the deflection or strain normal to x : 



W ■= W — Tt'j 

.*. d^w =: d^2u' — d^zUj^ ; 
hence, from Eqs. (9) and (12) : 



M _ dyu 
'EI~ ~d^ 



(13) 



Eq. (13) is exactly the same as Eq. (i) for straight beams. 

These investigations show that the common theory of flex- 
ure is not strictly applicable to the general case of curved 
beams. In order to obtain Eq. (12) it was necessary to assume 
the same law for stresses and strains, in any normal section, 
both for curved and straight beams, which is not exactly true. 
It was also necessary to assume small values of 

dw, . dw' 

~ and -^— 
dx dx 



Art. 1 8.] THE COMMON THEORY OF FLEXURE. 1 29 

for a close approximation. Yet the application of the common 
theory of flexure to curved beams, even within these restricted 
limits, is of the highest importance. 

In Art. 22 a generalization of the common theory of flexure 
is given, in which the differential of the centre line of the beam 
is used instead of dx. The resulting formulae are accurately 
applicable to curved beams of any curvature. The only as- 
sumption involved, in addition to those of the common theory, 
is the identity of the law of variation of stresses and strains in 
curved and straight beams ; and that causes very little error. 

One of the most important forms of Eq. (7) yet remains to 
be established. 

Let d^ represent the distance from the neutral axis of any 
normal section of the beam to that point of the section farthest 
from it. Let K represent the intensity of tensile or compres- 
sive stress (as the case may be) existing at this same point ; K 
will be the greatest intensity in the section. Because the in- 
tensity of stress varies directly as the distance from the neutral 
axis, the intensity at distance unity from that axis will be: 

d^ 

But by Art. 2, this intensity also has the value Eu, Con- 
sequently Eq. (7) becomes : 

KT 

^^=^ • • • (^4) 

If the external moment is sufficient to break the beam, and 
if Eq. (14) is applied to the section at which failure begins, K 
is called the ''Modulus of Rupture" for flexure. It is an 
empirical quantity. 



130 THEORY OF FLEXURE. [Art. 1 9. 



Art. 19. — Deflection by the Common Theory of Flexure. 

The common theory of flexure, as developed in the pre. 
ceding Art., leads to very simple and, in nearly all ordinary 
cases, very closely approximate formulae. 

Let Xq be the co-ordinate of some point at which the tan- 
gent of the inclination of the neutral surface to the axis of x 
is known ; then, from Eq. (i) of Art. 18 : 



dw 

dx 



r* M 



div 

—— will be at once recognized as the general value of the tan- 
gent of the inclination just mentioned, or, in the case of curved 
beams, as approximately the difference between the tangent, 
before and after flexure. 

Again, let x^ represent the co-ordinate of a point at which 
the deflection w is known, then, from Eq. (i) : 



w 



r^ M 

.-El'^' ...... (2) 

o 



The points of greatest or least deflection and greatest or 
least inclination of neutral surface are easily found by the aid 
of Eqs. (i) and (2). 

The point of greatest or least deflection is evidently found 
by putting : 

dw , . 

1^ = ° ^3) 



and solving for x. Since —J— is the value of the tangent of the 

dx 



Art. 19.] DEFLECTION. 131 

inclination of the neutral surface, it follows that a point of 
greatest or least deflection is found ivhere the beam is hori- 
zontal. 

Again, the point at which the inclination will be greatest or 
least is found by the equation : 



d 



fdw\ 



\ dx I d^w f V 

^ - -- o (4) 



dx dx"^ 



d^iv 
But, approximately, -j-^ is the reciprocal of the radius of 

curvature ; hence, the greatest inclination will be found at that 
point at which the radius of curvature becomes infinitely great, 
or, at that point at which the curvature cJianges front positive 
to negative or vice versa. These points are called points of 
" contra-flexure." Since : 

dx^ ' 



there is no bending at a point of contra-flexure. 

The moment of the external forces, M, will always be ex- 
pressed in terms of x. After the insertion of such values, Eqs. 
(i) and (2) may at once be integrated and (3) and (4) solved. 

The coefficient of elasticity, E, is always considered a con- 
stant quantity ; hence it may always be taken outside the in- 
tegral signs. In all ordinary cases, also, /is constant through- 
out the entire beam. In such cases, then, there will only need 
to be integrated the expressions : 



TX 



M dx and 



M dx\ 



-1 '-^0 



132 THEORY OF FLEXURE. [Art. 20. 

Before applying these formulas to particular cases it will be 
necessary to consider some other matters. 



Art. 20. — External Bending Moments and Shears in General. 

Beams subjected to combined bending and direct stress will 
not be treated. Such cases are of little or no real value to the 
engineer, and approximate solutions, even, are only to be 
reached by the higher processes of analysis. In all beams, 
therefore, pure bending only is to be treated. A beam is said 
to be non-continuo7LS if its extremities simply rest at each end 
of the span and suffer no constraint whatever, 

A beam is said to be continuous if its length is equal to two 
or more spans, or if its ends, in case of one span (or more) suffer 
constraint. 

A cantilever is a beam which overhangs its span ; one end of 
which is in no manner supported. Each of the overhanging 
portions of an open swing bridge is a cantilever truss. 

Let any beam be horizontal, and suppose it to be subjected 
to vertical loads. The results will evidently be applicable to 
any beam acted upon by loads normal to its axis. Let P be 
any single vertical load, and let x be any horizontal co-ordinate 
measured from any point as an origin. Let x\ represent the 
co-ordinate, measured from the same origin, of the point of ap- 
plication of any load P. Finally, let it be required to deter- 
mine the external bending moment J/ at any section, x, of the 
beam. The lever arm of any load P is evidently (x — x^. 

Hence, for any number of forces : 

M ^^P{x ^ x^ (i) 

The summation sign '2 refers only to x^ and is to cover that 
portion of the beam on one side of the section x^ as is evident 
from the manner of forming the equation. 



Art. 20.] MOMENTS AND SHEARS. 1 33 

If the origin of x is in the section considered : 

M ^ - :epx, . . . . e . , (2) 

From Eq. (i) : 

^J = ^^ = ^ (3) 

Now 2P= S is the algebraic sum of all the forces on one 
side of the section considered, it is consequently the total force 
acting in the section tending to move one portion of the beam past 
the other ; it is therefore called the '■^ shear'' in the section. 
This quantity (the shear) is a most important one in the sub- 
ject of the resistance of materials. 

The reactions, or supporting forces, applied to the beam, 
are to be included both in the sum ^P, and in the moment : 

^Pix-x^). 

Eq. (3) shows that the shear at any section is equal to the first 
differential coefficient of the bending moment considered as a 
function of x. 

The sum of all the loads on the other side of the section x 
would give the same numerical shear, but it would evidently 
have an opposite direction. 

As is well known, the analytical condition for a maximum 
or minimum bending moment in a beam is : 

dM 

^=^ • • . (4) 

From Eqs. (3) and (4) is to be deduced the following im- 



134 THEORY OF FLEXURE. [Art. 20. 

portant principle : The greatest or least bending moment in any 
beam is to be found in that section for which the shear is zero. 

The importance of this principle lies in the fact that in the 
greater portion of cases of loaded beams which come within 
the experience of the civil engineer, the section subjected to 
the greatest bending moment can thus be determined by a 
simple inspection of the loading. 

These principles can be well illustrated by the following 
simple example. 

Fig. I represents a non-continuous beam with the span /, 






i w-^ 



Fig. I. 

supporting two equal weights P, P. These two weights or 
loads are to be kept at a constant distance apart denoted 
by a. 

It is required to find that position of the two loads which 
will cause the greatest bending moment to exist in the beam, 
and the value of that moment. The reaction R is to be found 
by the simple principle of the lever. Its value will therefore 
be: 

R = S— ^ • 2P (5) 



Since the reaction R can never be equal to 2P, ^P, or the 
shear, it must be equal to zero at the point of application of one 
of the loads P. In searching for the greatest moment, then, 
it will only be necessary to find the moment about the point 



Art. 20.] 



MOMENTS AND SHEARS. 



135 



of application of one of the forces P, It will be most con- 
venient to take that one nearest R. 
The moment desired will be : 



M ^ Rx ^2P[x -^ -'!^ 

I 2/ 



• • (6) 



dM _ _ ^p f 2.r a 

cix X / 2/ 



/ a 

2 "" 4 



This value in Eq. (6) gives : 



Since : 



M. = ^[l 



dx"^ 



a + 



a' 
4/ 






(7) 



it appears that M^ is a maximum. 

If the load is uniformly continuous and of the intensity/, 

in Eqs. (i), (2) and (S)pdx^ is to be put forP, and the sign I for 

^. Hence : 



M =p {x- x^) dx,, 



M = -^ P 



x^ dx^ . 



136 THEORY OF FLEXURE. [Art. 20. 

dM [ , 



But since dx and dx^ are perfectly arbitrary, they may be 
taken equal to each other, hence : 



d^ 
dx 



—.^ =P' 



Or, the seco7td differential eoefficient of the vtoment^ considered 
as a function of x, is equal to the hitensity of the continuous 
load. 

A very important problem arises in connection with the 
principles discussed in this Art. It is the following : 

A continuous train of any given varyi^ig or uniform density 
advances alo7tg a simple beam of span I. It is required to deter- 
mine zvhat position of loading will give the greatest shear at any 
specified section. 

In Fig. 2, CD is the span /, and A is any section for which 
c A B D it is required to find the 

"^ ' ^^ Q position of the load for 

^^' ^ the greatest transverse 

shear. The load is supposed to advance continuously from C 
to any point B, Let R be the reaction at D, and ^P the load 
between A and B. The shear S' dX A will be : 

R-^P=. S' (8) 

Let R be that part of R which is due to :2P, and R' that 
part due to the load on CA ; evidently R is less than :2P, 
Then : 

R + R' - ^P ^ S'. 



Art. 20.] MOMENTS AND SHEARS. 137 

If AB carries no load, R and 'EP disappear in the value of 
S. Hence : 

R' = S 



is the shear for the head of the train at ^. 5 is greater than 
S' because 2P is greater than R. But no load can be taken 
from ^ (7 without decreasing i'?". Hence: The greatest shear 
at any section will exist when the load extends from the end of 
the span to that section, whatever be the density of the load. 

In general, the section will divide the span into two un- 
equal segments. The load also may approach from either 
direction. The greater or smaller segment, then, may be 
covered, and, according to the principle just established, either 
one of these conditions will give a maximum shear. A con- 
sideration of these conditions of loading in connection with 
Fig. 2, however, will show that these greatest shears will act in 
opposite directions. 

When the load covers the greater segment the shear is 
called a main shear; when it covers the smaller, it is called a 
counter shear. 

Again, let a continuous load of any given varying or con- 
stant density traverse a non-continuous beam ; it is required to 
find what position of the load will cause the greatest bending 
moment at any specified section. 

Since every load which can come upon the beam will pro- 
duce the same kind of bending at any given section, the greatest 
possible amount of load must be brought on the beam for the 
greatest bending moment at any section. 

Hence, tJie greatest bendijig moment , at the specified section, 
will exist when the load covers the whole span. 

It also follows that all sections will suffer their greatest 
bending moments with the same position of load. 

The principles involved in these results find important ap- 
plications in the theory of truss bridges. 



138 



THEORY OF FLEXURE. 



[Art. 21. 



Art. 21. — Moments and Shears in Special Cases. 

Certain special cases of beams are of such common occur- 
rence, and consequently of such importance, that a somewhat 
more detailed treatment than that already given may be 
deemed desirable. The following cases are of this character. 

Case I. 

Let a non-continuous beam, supporting a single weight P 

at any point, be considered, 
and let such a beam be rep- 
resented in Fig. I. If the 
span RR' is represented by 

I ^ a -^ b = RP-^ RP, 

the reactions R and R' will be : 




i? = ^ P, and R' = y P 



(I) 



Consequently, if x represents the distance of any section in 
RP from R, while x represents the distance of any section of 
RP from R , the general values of the bending moments for 
the two segments a and b of the beam will be : 

M = Rx, and M' = Rx' (2) 

These two moments become equal to each other and repre- 
sent f/ie greatest bendmg moment in the beam when 



X ■=^ a and x' = b, 



Art. 21.] 



SPECIAL MOMENTS AND SHEARS. 



139 



or, wJien the section is taken at the point of application of the 
load P. 

Eq. (2) shows that the moments vary directly as the dis- 
tances from the ends of the beam. Hence, if ^/'(normal to 
RR'^ is taken by any convenient scale to represent the greatest 

moment, — P, and if RAR is drawn, any intercept parallel to 

AP 2iVidi lying between RAR' and RR' will represent the bend- 
ing moment for the section at its foot, by the same scale. In 
this manner CD is the bending moment at D. 

The shear is uniform for each single segment ; it Is evi- 
dently equal to R for RP and R' for RP. It becomes zero at 
P, where is found the greatest bending moment. 

Case II, 



Again, let Fig. 2 represent the same beam shown in Fig. i, 
but let the load be one of uniform intensity, /, extending from 
end to end of the beam. Let C be placed at the centre of the 
span, and let R and R'y as be- a 

fore, represent the two reac- 
tions. Since the load is sym- 
metrical in reference to C, 

R = R'. 

For the same reason the mo- b " c> 

ments and shears in one half of the beam will be exactly like 
those in the other ; consequently, reference will be made to 
one half of the beam only. Let x and Xj then be measured 
from R toward C. The forces acting upon the beam are R 
and /, the latter being uniformly continuous. Applying the 
formulae of the preceding Art., the bending moment at any 
section x will be : 




140 THEORY OF FLEXURE. [Art. 21. 



M = Rx — p {x — X,) dx, . 



/. M=Rx^^ (3) 



If / is the span, at C, M becomes : 



2 8 

But because the load is uniform : 



R^^. 



^.^^-<^ (4) 



Hence ; 



^/.=%---^ ...... (5) 



if W is put for the total load. Placing 
in Eq. (3) : 



M =i- {Ix - X') o (6) 



2 



The moments M, therefore, are proportional to the abscis- 
sae of a parabola whose vertex is over C, and which passes 
through the origin of co-ordinates 7^. Let y^^T, then, normal 
to RR', be taken equal to M^, and let the parabola RAR be 



Art. 21.] SPECIAL MOMENTS AND SHEARS. I4I 



drawn. Intercepts, as FH, parallel to AC^ will represent bend- 
ing moments in the sections, as //", at their feet. 
The shear at any section is : 

. = « = .-,„,£_.) .. .<, 



or, it is equal to the load covering that portion of the beam be- 
tween the section i7i question and the centre. 

Eq. (7) shows that the shear at the centre is zero ; it also shows 
that S — R 2X the ends of the beam. It further demonstrates 
that the shear varies directly as the distance from the centre. 
Hence, take RB to represent R and draw BC. The shear at 
any section, as H^ will then be represented by the vertical in- 
tercept, as HG^ included between BC and RC. 

The shear being zero at the centre, the greatest bending 
moment will also be found at that point. This is also evident 
from inspection of the loading. 

Eq. (2) of Case /., shows that if a beam of span / carries a 

W 
weight — at its centre, the moment M at the same point will 

be: 

'.=T-hi^' («) 

The third member of Eq. (8) is identical with the third 
member of Eq. (5). It is shown, therefore, that a load, concen- 
trated at the centre of a non-continuous beam, will cause the same 
moment, at tJiat centre, as double the same load jcjtiformly dis- 
tributed over the span. 

Eqs. (5) and (8) are much used in connection with the bend- 
ing of ordinary non-continuous beams, whether solid or flanged : 
and such beams are frequently found. 



142 



THEORY OF FLEXURE. 



[Art. 21. 



Case III. 



The third case to be taken, Is a cantilever uniformly loaded ; 

it is shown in Fig. 3. Let x and x\ be 
measured from the free end A^ and let 
the uniform intensity of the load be 
represented by /. The entire loading 
is uniformly continuous. Hence the 
principles and formulae of Art. 20 give, 
for the moment about any section x : 




■ px'- 



M = — p {x — x^ dx^ = - 



. . (9) 



If AB = /, the moment at ^ is : 



_ //^ 



2 



(10) 



The negative sign is used to indicate that the lower side of 
the beam is subjected to compression. In the two preceding 
cases, evidently, the upper side is in compression. 

The shear at any section is : 



S = = — px 

ax 



(II) 



Hence, the shear at any section is the load between the free 
end and that section. 

Eq. (9) shows that the moments vary as the square of the 
distance from the free end ; consequently, the moment curve 
is a parabola with the vertex at A, and with a vertical axis. 
Let BC, then, represent M^ by any convenient scale, and draw 



Art. 22.] 



GENERAL FLEXURE FORMULAE. 



H3 



the parabola CD A, Any vertical intercept as DFv^'iW. repre- 
sent the moment at the section, as F, at its foot. 

Again, let BG represent the shear//, at B^ then draw the 
straight line AG. Any vertical intercept, as HF^ will then 
represent the shear at the corresponding section F, 



Art. 22. — Recapitulation of the General Formulse of the Common 

Theory of Flexure. 

It is convenient for many purposes to arrange the formulae 
of the Common Theory of Flexure in the most general and 
concise form. In this Art. the preceding general formulae for 
shears, strains, resisting moments and deflections will be re- 
capitulated and so arranged. In order to complete the gener- 
alization, the summation sign '2 will be used instead of the 
sign of integration. 




In Fig. I, let ABC represent the centre line of any bent 
beam ; AF^ a vertical line through A ; CF^ a horizontal line 
through Cy while A is the section of the beam at which the 
deflection (vertical or horizontal) in reference to Cy the bend- 
ing moment, the shearing stress, etc., are to be determined. 
As shown in figure, let x be the horizontal co-ordinate meas- 
ured from A^ andjK the vertical one measured from the same 
point ; then let z be the horizontal distance from the same 



144 THEORY OF FLEXURE. [Art. 22. 

point to the point of application of any external vertical force 
P. To complete the notation, let D be the deflection desired ; 
M^y the moment of the external forces about A ; vS, the shear 
at A ; P\, the strain (extension or compression) per unit of 
length of a fibre parallel to the neutral surface and situated at 
a normal distance of unity from it ; /, the general expression of 
the moment of inertia of a normal cross section of the beam, 
taken in reference to the neutral axis of that section ; E^ the 
coefficient of elasticity for the material of the beam ; and M 
the moment of the external forces for any section, as B. 

Again, let A be an indefinitely small portion of any normal 
cross section of the beam, and let y' be an ordinate normal 
to the neutral axis of the same section. By the *' common 
theory " of flexure, the intensity of stress at the distance y' 
from the neutral surface is (y'P'E). Consequently the stress 
developed in the portion A, of the section, is EP'y A, and the 
resisting moment of that stress is EP'y'^A. 

The resisting moment of the whole section will therefore 
be found by taking the sum of all such moments for its whole 
area. 

Hence : 

M = EP"2y'^A = EP'L 
Hence, also . 

EI' 

If n represents an indefinitely short portion of the neutral 
surface, the strain for such a length of fibre at unit's distance 
from that surface will be itP'. 

If the beam were originally straight and horizontal, n would 
be equal to dx. 

P' being supposed small, the efiect of the strain fiP' at any 



Art. 22.] GENERAL FLEXURE FORMULAE. 145 

section, B, is to cause the end K of the tangent BK, to move 
vertically through the distance nP'x, 

If BK and BR (taken equal) are the positions of the tan- 
gents before and after flexure, nP' x will be the vertical dis- 
tance between K and R. 

By precisely the same kinematical principle, the expres- 
sion nP'ywWi be the horizontal movement oi A in reference 
to B, 

Let ^itP'x and '^nP'y represent summations extending 
from A to C, then will those expressions be the vertical and 
horizontal deflections, respectively, of A in reference to C, It 
is evident that these operations are perfectly general, and that 
X and y may be taken in any direction whatever. 

The following general, but strictly approximate equations, 
relating to the subject of flexure, may now be written : 




D^ represents horizontal deflection. 

10 



146 THEORY OF FLEXURE. [Art. 23. 

The summation ^Pz must extend from ^4 to a point of no 
bending ; or from ^ to a point at which the bending moment 
is M^, In the latter case : 

M, = 2Pz -{-M/ (7) 

J// may be positive or negative. 

Art. 23. — The Theorem of Three Moments. 

The object of this theorem is the determination of the re- 
lation existing between the bending moments which are found 
in any continuous beam at any three adjacent points of sup- 
port. In the most general case to which the theorem applies, 
the section of the beam is supposed to be variable, the points 
of support are not supposed to be in the same level, and at 
any point, or all points, of support there may be constraint 
applied to the beam external to the load which it is to carry ; 
or, what is equivalent to the last condition, the beam may not 
be straight at any point of support before flexure takes place. 

Before establishing the theorem itself, some preliminary 
matters must receive attention. 

If a beam is simply supported at each end, the reactions 
are found by dividing the applied loads according to the 
simple principle of the lever. If, however, either or both ends 
are not simply supported, the reaction, in general, is greater at 
one end and less at the other than would be found by the law 
of the lever ; a portion of the reaction at one end is, as it were, 
transferred to the other. The transference can only be ac- 
complished by the application of a couple to the beam, the 
forces of the couple being applied at the two adjacent points 
of support ; the span, consequently, will be the lever arm of 
the couple. The existence of equilibrium requires the appli- 



Art. 23.] 



THEOREM OF THREE MOMENTS. 



H7 



cation to the beam of an equal and opposite couple. It is only 
necessary, however, to consider, in connection with the span 
AB, the one shown in Fig. i. Further, from what has imme- 
diately preceded, it appears that the force of this couple is 




equal to the difference between the actual reaction at either 
point of support and that found by the law of the lever. The 
bending caused by this couple will evidently be of an opposite 
kind to that existing in a beam simply supported at each end. 
These results are represented graphically in Fig. i. A and 
B are points of support, and AB is the beam ; AR and BR' 
are the reactions according to the law of the lever ; RF = R'F 
is the force of the applied couple ; consequently : 

AF^ AR-\- RF and BF = BR' - {R'F = RF) 



are the reactions after the couple is applied. As is well known, 
lines parallel to CK, drawn in the triangle ACB, represent the 
bending moments at the various sections of the beam, when 
the reactions are AR and BR. Finally, vertical lines parallel 
to AG, in the triangle QHG, will represent the bending mo- 
ments caused by the force R'F. 



148 THEORY OF FLEXURE. [Art. 23. 

In the general case there may also be applied to the beam 
two equal and opposite couples, having axes passing through 
A and B respectively. The effect of such couples will be 
nothing so far as the reactions are concerned, but they will 
cause uniform bending between A and B. This uniform or 
constant moment may be represented by vertical lines drawn 
parallel to AH ox ZtV (equal to each other) between the lines 
AB and HQ. The resultant moments to which the various 
sections of the beam are subjected will- then be represented by 
the algebraic sum of the three vertical ordinates included be- 
tween the lines ACB and GQ. Let that resultant be called M, 

Let the moment GA be called M^, and the moment : 

BQ = LN = HA = M,. 

Also designate the moment caused by the load P, shown by 
lines parallel to CK in A CB, by M^. Then let x be any hori- 
zontal distance measured from A toward B ; / the horizontal 
distance AB ; and z the distance of the point of application, 
/v", of the force P from A, With this notation there can be at 
once written : 

M = Mf--^) + ^,g) + J/, .... (I) 



Eq. (i) is simply the general form of Eq. (2). 

It is to be noticed that Fig. i does not show all the mo- 
ments M^, Ml, and J/^ to be of the same sign, but, for conven- 
ience, they are so written in Eq. (i). 

The formula which represents the theorem of three mo- 
ments can now be written without difficulty. The method to 
be followed involves the improvements added by Prof. H. T. 
Eddy, and is the same as that given by him in the ''American 
Journal of Mathematics," Vol. I., No. i. 



Art. 23.] 



THEOREM OF THREE MOMENTS. 



149 



Fig. 2 shows a portion of a continuous beam, including two 
spans and three points of support. The deflections will be 
supposed measured from the horizontal line NQ. The spans 



■^- 






1 — 



14 z ^ 

"7X 



s s. 



IC. 






^ N- 



V 




Fig. 2 



are represented by 4 and 4; the vertical distances of NQ from 
the points of support by <f^, Cj, and <:^ ; the moments at the same 
points by M^, M^, and M^, while the letters 5 and R represent 
shears and reactions respectively. 

In order to make the case general, it will be supposed that 
the beam is curved in a vertical plane, and has an elbow at b, 
before flexure, and that, at that point of support, the tangent 
of its inclination to a horizontal line, toward the span 4 is t, 
while t' represents the tangent on the other side of the same 
point of support ; also let d and d' be the vertical distances, 
before bending takes place, of the points a and c, respectively, 
below the tangents at the point b. 

A portion of the difference between c^. and <r^ is due to the 
original inclination, whose tangent is 4 and the original lack of 
straightness, and is not caused by the bending ; that portion 
which is due to the bending, however, is, remembering Eq. 
(5), Art. 22 : 



D = c^- Ci, - IJ - d = 2^ 



« Mxft 



EI 



ISO 



THEORY OF FLEXURE. 



[Art. 23. 



By the aid of Eq. (i) this equation may be written : 



E {c^ - c^ - I J - d) 



= ^, 



M. 



X 



+ M,{-^ ) + M, 



xn 



. (2) 



In this equation, it is to be remembered, both x and z (in- 
volved in M^ are measured from support a toward support b. 
Now let a similar equation be written for the span 4, in which 
the variables x and z will be measured from c toward b. There 
will then result : 

E (r, - cj,- 1/ - d') • 



= 2 



M. 



I- X 



+ J/,(7) + ^4tJ 



. (3) 



When the general sign of summation is displaced by the 
integral sign, n becomes the differential of the axis of the 
beam, or ds. But ds may be represented by u dx, u being such 
a function of x as becomes unity if the axis of the beam is 
originally straight and parallel to the axis of ;r. The Eqs. (2) 
and (3) may then be reduced to simpler forms by the following 
methods : 

In Eq. (2) put : 



I — X 



xn 



I 

U 



'" u (4 — x)x dx 



a'' b 



I 



x^ V" u (4 — ;r) dx 

t^a. J h J- 



(4) 



Also : 



Art. 23.] THEOREM OF THREE MOMENTS. 



151 






^ U (4 — ^) dx _ iaXa 

I ~ L 



u{la - x)dx . . . (5) 



Also: 



'-^1 ./(4-^)^^ 



^a J^f 






{lj-x)dx = ^^^ (6) 



In the same manner : 



« ;r^;/ _ I 
^77" ~ 7 J 



'^ iL^y" dx _ x'a 



'^ iix dx 



. (7) 



Also: 



X 



* 2^;jr dx 



L 



ux dx 



(8) 



And, 



^/7 -^/l 



'^n^ nU. 



UX dx = '^ J '^ 



xdx = Mi?ff^ 
2 



• (9) 



Again, in the same manner : 



M^xn 



I 



— ~ i^ji^a^M^x Ax . ... (10) 



Using Eqs. (4) to (10), Eq. {2) may be written : 



E{c, - c- IJ - d) = ^ {MjiJ^x, + M,u'jy:) 



+ U^aha^^Mx Ax 



(II) 



152 THEORY OF FLEXURE. [Art. 23. 

Proceeding in precisely the same manner with the span 4, 
Eq. (3) becomes : 



+ uJ,,^JI,xAx (12) 

The quantities x^ and x^ are to be determined by applying 
Eq. (4) to the span indicated by the subscript ; while u^j ia, u^ 
and ic are to be determined by using Eqs. (5) and (6) in the 
same way. Similar observations apply to m'^, t^, x^, t/„ i'^ and 
x'^y taken in connection with Eqs. (7), (8) and (9). 

If / is not a continuous function of x, the various integra- 
tions of Eqs. (4), (5), (7) and (8) must give place to summations 
(^) taken between the proper limits. 

Dividing Eqs. (11) and (12) by 4 ^i^d 4, respectively, and 
adding the results : 

77 f ^a ^b [ ^c ^b 7- " ^ 



= 'i^ 2\M,x Ax + '!^ 2\ M,x Ax 

'■a ^c 

+ y2{MjiJ^x^ + Mt,u'jy^ + M.uJ^v, + M^u\i'X) . (13) 

in which T — t -\- 1\ 

Eq. (13) is the most general form of the theorem of three 
moments if E, the coefficient of elasticity, is a constant quan- 
tity. Indeed, that equation expresses, as it stands, the ** the- 
orem " for a variable coefficient of elasticity if (ie) be written 
instead of i\ e representing a quantity determined in a manner 
exactly similar to that used in connection with the quantity L 



Art. 23.] THEOREM OF THREE MOMENTS. 1 53 

In the ordinary case of an engineer's experience 7" = o, 
d == d' =^ o, / = cojtstant, u = ti^ = 21^ = ctc.^ ^=z c ^ secant of 
the inclination for which t = — t' is the tangent ; consequently : 



2/. 

Xc = —z- 



From 


Eq. (4) : 








Xa 


= 


24 

6 ' 


From 


Eq. (7) : 








K 


= 


44 
6 ' 



X - ^ 

'~ 6 



The summation 2M^x Ax can be readily made by referring 
to Fig. I. 

The moment represented by CK in that figure is : 



I 



consequently the moment at any point between A and Ky 
due to P, is : 



Between K and B : 



^.'= (ttt^)- CK=p'j{l-x). 



154 THEORY OF FLEXURE. [Art. 23. 

Using these quantities for the span 4 • 

^\m^x Ax = \ M^x dx + f M^x dx = yiP{/^ - z')z. 



For the span 4, the subscript a is to be changed to c. 
Introducing all these quantities Eq. (13) becomes, after 
providing for any number of weights, P: 

-4-- ( ''^ 7 '^ + ''~/' ) - ^^Ja + 2iF,(4 + /.) + MJ, 



+ J ^P{i! - zy + 1 hp{i! - ->- . . . (14) 



Eq. (14), with c' equal to unity, is the form in which the 
theorem of three moments is usually given ; with c' equal to 
unity or not, it applies only to a beam which is straight before 
flexure, since : 

T=t-\-f = o = d^d'. 

If such a beam rests on the supports ^, b, and r, before 
bending takes place, 

Ca — Cu €,. — Cf, 



la Ic ' 

and the first member of Eq. (14) becomes zero. 

If, in the general case to which Eq. (13) applies, the deflec- 
tions c^y Cb, and c, belong to the beam in a position of no bend- 
ing, the first member of that equation disappears, since it is 
the sum of the deflections due to bending only, for the spans 4, 
and 4 divided by those spans, and each of those quantities is 



Art. 23.] THEOREM OF THREE MOMENTS, 155 

zero by the equation immediately preceding, Eq. (2). Also, if 
the beam or truss belonging to each span is straight between 
the points of support {such points being supposed in the same 
level or 7zot)^ //^ = ti'^ = 11^^ = constant, and z/^ = u'^ = u^^ = a7i- 
other constant. If, finally, /be again taken as constant, x^ and 
Xcy as well as '^M^x Ax, will have the values found above. 

From these considerations it at once follows that the second 
member of Eq. (14), put equal to zero, expresses the theorem 
of three moments for a beam or truss straight between points 
of support, when those points are not in the same level, but* 
when they belong to a configuration of no bending in the 
beam. Such an equation, however, does not belong to a beam 
not straight between points of support. 

The shear at either end of any span, as 4, is the next to be 
found, and it can be at once written by referring to the obser- 
vations made in connection with Fig. i. It was there seen 
that the reaction found by the simple law of the lever is to be 
increased or decreased for the continuous beam, by an amount 
found by dividing the difference of the moments at the ex- 
tremities of any span by the span itself. Referring therefore, 
to Fig. 2, for the shears 5, there may at once be written : 



^a — ^^ -J ■ -, ■ . . . . (,I3J 



5.= ipf + ^-7^- . .... (:6) 



I'C h 




15^ THEORY OF FLEXURE. [Art. 23. 

The negative sign is put before the fraction, 

4 

in Eq. (15), because in Fig. i the moments M^ and M^ are rep- 
resented opposite in sign to that caused by P^ while in Eq. (i) 
the three moments are given the same sign, as has aheady 
been noticed. 

Eqs. (15) to (18) are so written as to make an upward re- 
action positive, and they may, perhaps, be more simply found 
by taking moments about either end of a span. For example, 
taking moments about the right end of 4 • 

5X - ^P{la - ^) + M^ = M,, 

From this, Eq. (15) at once results. Again, moments about 
the left end of the same span give : 

This equation gives Eq. (16), and the same process will give 
the others. 

If the loading over the different spans is of uniform inten- 
sity, then, in general, P=wd2; w being the intensity. Con- 
sequently : 

[I h 

:EP{1^ - z^)z = IV (/^ - jj^::^dz=w — , 
K 4 

In all equations, therefore, for 

j-2P{a-^^)^ 



Art. 23^.] CONTINUOUS REACTIONS. 1 5/ 

there is to be placed the term Wc, -^ ; and for 

4 



-1 i/^(// - ^0 ^ 



/3 

the term w^ -^. The letters a and ^ mean, of course, that 
4 

reference is made to the spans 4 ^^^d 4. 

From Fig. 2, there may at once be written : 



R =S:Ar S^ (19) 

R ^ S,-^ S, (20) 

R' ^S:^S, (21) 

etc. = etc. + etc. 

Art. 23a. — Reactions under Continuous Beam of any Number of Spans. 

The general value of the reactions at the points of support 
under any continuous beam have been given in Eqs. (19), (20), 
(21), etc., of the preceding Art. Before those equations, how- 
ever, can be applied to any particular case, the values of the 
bending moments, which appear in the expressions S^, 5^', Sj,y 
etc., for the shears, must be determined. In the application of 
the theorem of three moments, it is invariably virtually as- 
sumed that the continuous beam before flexure is straight 
between the points of support, and that the latter belong to a 
configuration of no bending. The moment of inertia / and 
the coefficient of elasticity E are also assumed to be constant. 
This is frequently not strictly true, yet it will be assumed in 



15B THEORY OF FLEXURE. ' [Art. 23^. 

what follows, since the method to be used In finding the mo- 
ments is entirely independent of the assumption, and remains 
precisely the same whatever form for the theorem of three 
moments may be chosen. 

Agreeably to the assumption made, Eq. (14) of the preced- 
ing Art. takes the following form, which is almost, or quite, 
invariably used in engineering practice : 






- ^ ^P{1} -z')z (I) 

Let Fig. I represent a continuous beam of n spans, equal 
or unequal in length. At the points of support, o, i, 2, 3, 4, 5, 



^2 ^3 A ^ 5 U 

T '^ ^ ^ ^ ^ 

Fig.l 



etc., let the bending moments be represented by M^, M^, M^y 
My etc. The moment M^ is always known ; it is ordinarily 
zero, and that will be considered its value. 

An examination of Fig. i shows that, by repeated applica- 
tions of Eq. (i), the number of resulting equations of condition 
will be one less than the number of spans. But if the two end 
moments are known (here assumed to be zero), the number of 
unknown moments will also be one less than the number of 
spans. Hence the number of equations will always be sufB- 
cient for the determination of the unknown moments. 

For the sake of brevity let the following notation be 
adopted : 



Art. 23<^.] CONTINUOUS REACTIONS. 1 59 

2/, = - -j- kP{l,^ - ^)Z - -1 i'/^(// - 2^)2 , 

?/, = - -^ i"'/'(4' - z')^ - ~ ^PiJ^ - ->. 

etc. = etc. — etc. 



a, = /,; b^ = 2(4 + 4) ; c^ ^ I^. 

^3 = 4; ^3 == 2(4 4- 4) ; 4 = 4- 
^4 = 4; 4 = 2(4 + 4) ; /4 = 4 • 

A- = 4 ; ^.- = 2(4 + 4 , ,) ; Si = li.^', 

i denoting any number of the series i, 2, 3, 4, n. It is 

thus seen that, in general, 

qi - 2(A- + Si) ; 

also that a^ = b^, c^ = by d^ — c^, etc. These relations can be 
used to simplify the final result. 

By repeated applications of Eq. (i) the following n equa- 
tions of condition, involving the notation given above, will 
result : 



i6o 



THE OR Y OF FLEXURE. ' 



[Art. 23^. 






= ?/, 



= ?/. 



= ^/o 



u. 



= 2/. 



(2) 



= u^ 



The moment J/« + , will also be equal to zero. In conse- 
quence of this last condition it is seen that the coefficients of 
the Ms occupy precisely the places of the elements of a deter- 
minant of the n^^ degree. Of the array indicating the deter- 
minant, however, there exists only the leading diagonal and 
one diagonal on each side of it. The determinant for n equa- 
tions, or (;/ + i) spans, has, then, the value : 

^i, /?!, o, o , o, o , 1 

a^, l?2, c^y o , o , o , 

0,^73,^3,^^3,0,0, 

^ = ^ o , o , r^, ^4, /,, o , )■ • • • (3) 

0,0,0, ^5, 7, ^5, 

• • • • • 

.0,0,0,0,0, . . . . o, /„, ^« 
Also let Di represent the value of the determinant D when 



Art. 23<^.] 



CONTINUOUS REACTIONS. 



l6l 



the column indicated by the i^^ letter of the series ^, b^ c^ d, /", 
etc., is replaced by the column ti^, ti^j iiy u^j etc. If, for exam- 
ple, t = 3, the t^^ letter is c. Hence : 



' ^n ^ij ^^» 0,0,0, 

^2, <^2, 7/^, 0,0,0,. . . . . 

O, by Uy dyOyO, 

O jOy 21^, d^,/^,0, 1- • • • (4) 

0,0, 2^5, d^, /5, ^5, 



n. = 



O , O , 7/„, O , O , O , . . O, /«, ^„ J 



Then, in general : 



M,= 



D 



(5) 



Eq. (5) will give the value of the bending moment at any 
point of support, whatever may be the number of spans or the 
law of loading on any or all the spans. 

Precisely the same formulae are to be used if il^o ^^id M^ are 
not zero, but have definite values and are known. In such a 
case, however, 11^^ and 2/„ would be replaced by : 

u\ = n, — a^M^. 

The same equations also hold true whatever form of the 
theorem of three moments may be chosen. It is only to be 
II 



h 



1 62 THEORY OF FLEXURE. [Art. 23^. 

remembered that the values of the quantities a^ b, c, etc., z/j, 
ti^, ti^j etc., will depend upon the choice. 

If all the moments are desired, it will be most convenient 
to put the vertical column it^, 71^, Jiy . . , ti^ in place of the ver- 
tical column ^j, a^, o, o, . . . o, in Eq. (4), and then find the 
resulting determinant D^. Eq. (5) will then give the value of 
Jfj, which, placed in the first of Eqs. (2), will enable M^ to be 
at once found. M^ will then result from the second of Eqs. (2), 
M^ from the third, etc., etc. 

So far as the general treatment of the question is con- 
cerned, there yet remains to be considered the expansion of 
the determinants D and Di. 

The expansion of the determinant D is very simple, and 
leads to the following results : 

For two spans : 

D = a^ (6) 

For'three spans : 

D — aj?^ — aj)^ ...-,... (7) 
For four spans : 

D = ajb^c^ — aj)^c^ — ajb^c^ (8) 

For five spans : 
D — ajb^c^d^ — ajj^c^d^ — ajb^c^d^ — ajj^c^d^ -f ciJb^Cj^d^ . (9) 

For six spans : 
D - aj?^c^dj^ - ajj^c^dj^ - aj\c^dj^ — aji^c^dj^ + aj)^c^d^f^ 
- aj?^c^dj^ + ajj^c^dj^ + ajj^c^dj^ . . . (10) 



Art. 23^;.] CONTINUOUS REACTIONS. 1 63 

By the observance of two or three simple rules, the deter- 
minant for (11 Ar i) spans, or n points of support, may easily be 
written. 

A series of numbers such as I, 2, 3, 4, 5> 6, etc., is said to 
be written in its natural order. Let any permutation of this 
series, 2, i, 3, 6, 5,4, be written, in which 2 is placed before i, 
6 before 5 and 4, and 5 before 4. In this permutation, there- 
fore, there are said tobe(i+2-j-i)=4 inversions. 

Let (A,J represent any letter of the series a^ b, c, d, etc., Avhich 
has the subscript n\ also, let (/\„)„ and (AJ„_j represent the 7& 
and {11 — if^ letters of the same series which have the sub- 
scripts n. In general, the letter inside the parenthesis repre- 
sents the subscript figure in the determinant, and that outside, 
the place of the letter in the series a, b^ c, d, f, etc. 

The n*^ determinant for {n -\- i) spans, of ;/ points of sup- 
port, will then be : 

Now, with the notation taken, if the letters in each term of 
the determinant are written in their natural order, as abcdfg, 
etc., the niunber of inversions in the subscript figures of any term 
will determine the sign of that term, i.e., if the number of in- 
versions is odd, the sign is minus, but if the number is even the 
sign is plus. 

Since n is the greatest subscript in any term, and since (A^)^ 
occupies the most advanced place in the series of letters, no 
inversions are introduced in multiplying D^_^ by (A„)„. Hence, 
all terms of D„_^ (A„)„ zuill have the same signs as the correspond- 
ing terms of D„_^. 

Similarly, since n is greater than {jt — i), the product 
{^^7i-i{^n-iin involves one inversion. Hence, all terms of 

■'-^n - 2 \ njn - i V m - i '« 



164 



THEORY OF FLEXURE. 



[Art. 2ia. 



will Jiave signs contrary to those of the corresponding terms of 

The number of terms in D^ will evidently be the sum of 
the numbers of terms in i^«_i and D^_^. 

An examination of the notation will at once show that : 

{\X = 2(4 + 4 + 1) ; {^r^i-i = 4 ; and {K-z)n = 4. 

Hence there will result : 

-^« = 2i^„_,(4 + 4.x) - A-.4. . . . (II) 

The minus sign before the last term of the second member 
is on account of the inversion introduced, as already ex- 
plained. 

The general value of the determinant Di (shown in Eq. (4) 
when i = 3) can be most easily expanded by considering it 
the sum of two determinants ; and in order to illustrate this 
method let it be supposed that M^ is desired. It will then be 
necessary to expand the determinant D^, given in Eq. (4). As 
is known from the theory of determinants, D^ may be written 
as follows : 



V. = 



'a„ b,,o ,0 ,0,0 , 
a^, b^, 21^, 0,0,0, 
o , ^3, ziy 4' o , o , 
0,0, u^, d^, f, o , 
o , o , o , ^5, /5, ^5, 



^a^, b^, ?/„ 0,0,0, . 

<72, (^2, O , O , O , O , . 

o , by o , dy o , o , . 

o , o , o , ^4, Z^, o , . 
o o 11 d f o- 



h + ^ 



O, O, O, O, O, 0,0,.. />„, qj lo, O, u^, O, 0,0,.. /„, q^ J 



KI2) 



Art. 23^.] CONTINUOUS REACTIONS. 1 65 

or : 

i?3 = z); + 2>3" (13) 

Eq. (12) shows at a glance what D^ and D^' represent. 

D^ is precisely the same in form as D, and is given at once 
by the Eqs. (6) to (11) after writing ii^, u^ and ii^ for c^^ c^ 
and c^. 

In general, D/ is found by simply writing tCi.^, Ui and z/, + i 
for (/\_iX-, (A,),- and (A^ + i) »• in the determinant D. 

As a general method, that of alternate numbers is probably 
as simple as any for the expansion of the determinant Di' . 
For example : 

in which r^ ^2, ^3, etc., are the units of the alternate numbers. 

The circumstances of any particular case will frequently 
either furnish a more expeditious method than that of alternate 
numbers, or allow the expansion of D-' to be written at once 
from an inspection of the array given in Eq. (12). 

In any case the method of alternate numbers may be used 
as a check. 

Special Method for Ordinary Use, 

If the number of spans is large, the expansion of the deter- 
minant Di will, at best, be found somewhat tedious. Special 
methods may be employed which involve only the determinant 
/>, given in Eqs. (6) to (11) ; and it has already been seen that 
that determinant admits of a very simple expansion. 

Let any one span carry any load whatever^ while all other 
spans carry no load. In such a case, P will be zero for every 



1 66 THEORY OF FLEXURE, [Art. 23«. 



Span but one, and, in consequence of the notation employed, 
all but two quantities in the series u^^ u^, Uy u^, u^ etc., will also 
become equal to zero. 

If li (the i^^ span) carries the load, there will result : 



U,.,:=. -^^P{ll -Z^)Z (15) 



li 



u, = - 4 -^^(^^ - ^> (i^) 



h 



All other ?/s reduce to zero. Although Eqs. (15) and (16) 
have the same form, they are not identical except in special 
cases, since z is not measured from the same end of the span 
in both expressions. 

Now let 2/j_i and Ui take the place of those letters in that 
column of D formed with the i^^ letter of the series a, b^ c, d, 
etc., which have the subscripts i and i — i ; Ui^^ is equal to 
zero. Or in the notation already employed, let z/j_i and z/^ 
take the place of (A,-_i),- and (/\),-, while zero takes the place of 
(A,- + i),-. The resulting determinant, D^, will then be precisely 
the same as D in general form. The expansion of Di can then 
be at once made by simply putting in D the substitutions 
above indicated. There will then result : 



^■ = §' (17) 



In order to find Mi_^, with the same loading on the same 
span, tii_^ and Ui must take the place of (A;_j\_j and (A,),-i, re- 
spectively, while (A,_2\-_i becomes equal to zero. Making 
these substitutions in the determinant D, there will result the 
determinant Z^,_ I- Then: 



Art. 23a.] CONTINUOUS REACTIONS. 167 

^.-.-%^ (18) 



The values of Mi and Mi_^, thus obtained, placed in the ^"^^ 
and {i — i)'^ of the Eqs. (2) will at once give ; 

Mi.^ and J/,-^,. 

Similar substitutions in the other equations will give all the 
moments. Thus the solution is complete, for the span and 
loading taken, with the use of the expanded determinant D 
only. 

Each span may be treated in the same manner and the 
same expansion of D will be the only one necessary. 

This method is equivalent to splitting the elements ?/„ ii^^ 
tiy u^, etc., of the general determinant Di. 

In order to determine the bending moment at any point of 
support, for loading which covers more than one span, or por- 
tions of more than one span, it is only necessary to take the 
algebraic sum of the separate moments (as above determined), 
at the point of support in question, found for the loading in 
each single span. The result will 'be the moment due to the 
combined action of all the loading. 

It is thus seen that the solution of the most general case is 
made to depend on the one expansion of the determinant D, 



Example, 

Let there be a continuous beam of six spans, and let any 
loading rest upon the fourth ; it is required to find the expan- 
sions of the determinants Di and Di_^. 

The expansion of D is given in Eq. (10), and need not be 
repeated here. 



l68 THEORY OF FLEXURE. [Art. 23^. 

Using the preceding notation : 

i =4. (A._^). = ^3. 



^.•-i = ?V 



In Eq. (10) then, d^ and d^ are to be displaced by u^ and 2/3, 
while zero is to take the place of d^. Hence : 

+ «2<^i^4^3/5 (19) 

Again : 

Then, in Eq. (10), placing u^^ and ii^ for c^ and ^4, and placing 

^i-^i-'L = ^2 = 0, 
there will result : 

A- I = ^i^2?^3^4/5 ~ ^2'^i?^3^4/5 ~ ^J^^^U^zfs + ^2^i^U^J''> 

— ajj^ii^d^f^ + aj)^u^d^f^ (20) 

These values placed in Eqs. (17) and (18) will give M^ 
and 3/3. 

The lengths of span maybe any whatever ; if they are equal, 
the results will be simplified. 



Art. 23^.] CONTINUOUS REACTIONS. 1 69 

Special Case of Equal Spans. 

If all the spans are of equal length, each may be repre- 
sented by /. There will then result : 

a^ — bj^^c^ — d^—... =p. — b^ — c^ — d^ =f^ = , , . Si= I) 

\ (21) 
a, = b, = c^ = d^ =f = , . . =qi = 4l, ) 

These values of a, b, c, etc., placed in Eqs. (6) to (10) give : 

For two spans : 

D =4/. 

For three spans ; 

D ^ isl\ 

For four spans : 

n = 56/3. 

For five spans : 

D = 209/4. 
For six spans : 

D = 780/5. 

Others may be easily and rapidly written by the aid of Eq. 
(11), which now becomes : 

D„ = 4lD„_,- PD^., (22) 

If the determinant for seven (z.e.y n -\- i) spans is desired : 



I70 THEORY OF FLEXURE. [Art. 2^a, 

D„_^ = 780/5 and Z>„ _ 2 = 209/ 4. 
Hence : 

B^ = Dq = 3120/^ — 209/^ = 291 1/^ 

Similarly for eight spans : 

D = 4 X 291 1/7 - 780/7 = 10864/7. 

For nine spans : 

D = 4. X 10864/^ — 2911/^ = 40545/^ 

For ten spans : 

Z> = 4 X 40545/9 — 10864/9 = 15 13 16/9. 

The values given in Eq. (21) will correspondingly simplify 
the expansion of the determinant i^,-, either in its general form 
as exemplified in Eq. (4) or as given in the special method. 
As an illustration, Eqs. (19) and (20) become, respectively : 

Di_^ = 225/^3 — 60/ *u^. 
These values then give : 

''''- D - 52/ • 



Art. 24.] THE NEUTRAL CURVE. I /I 

Then by Eqs. (2) : 

M, = "f - (4 J/3 + M,) . 



M,= 'ji-(M, + 4M,) = -^ 



2 



Thus all the moments are known for this example, />, with 
six spans and loading on the fourth span only. 

Rcactiojzs. 

After the moments are found either by the general or 
special fnethod, for any condition of loading, the reactions will 
at once result from the substitution of the values thus found 
in the Eqs. (15) to (21) of the preceding Art., which it is not 
necessary to reproduce here. 



Art. 24. — The Neutral Curve for Special Cases. 

The curved intersection of the neutral surface with a ver- 
tical plane passing through the axis of a loaded, and originally 
straight, beam may be called the " neutral curve." The neu- 
tral curve is the locus of the extremities of the ordinates w of 
Art. 19; it therefore gives the deflection at any point of the 
beam. ' 

The method of finding the neutral curve for any particular 
case of beam or loading can be well illustrated by the opera- 
tions in the following three cases. 



172 



'theor y of flexure. 



[Art. 24. 



I 



Case L 

This case is shown in the accompanying figure, which 

represents a cantilever carry- 
ik \ ing a uniform load with a sin- 

/ k— ^^ — >i gle weight W^at its free end. 

B As usual, the intensity of the 

uniform loading will be repre- 

i., sented by p. 

Fig.l j Measuring x and w from 

' B^ as shown, the general value 



£ 



of the bending moment is : 



dx' 2 



• (I) 



Integrating between x and /, remembering that : 

dw 



for X = I : 



dx 



o 



dw W , . ,„. . / 



^^^ = ^(-^-^o+-&(^'-^') . • • w 



^;ir 



6 



Hence : 



"=i7{f(f-'")+ie--)[ ■■<^' 



The greatest deflection, «-„ occurs for x — I. Hence : 



Art. 24.] THE NEUTRAL CURVE. 173 

The greatest moment, J/j, exists at A^ and its value is : 



M,= W/ + ^ (5) 



These equations are made applicable to a cantilever with 
a uniform load by simply making W — o. They then be- 
come : 



= EI —r = -— 



^-^^^.-r = V («) 



^^J=^^'-^') (7) 



- = 6-17(7-''^) (') 






'2 



M, = ^ (10) 

2 ^ ^ 



Again, for a cantilever with a single weight only at its free 
end,/ is to be made equal to zero in the first set of equations. 
Those equations then become : 

M = EI^%-= Wx (II) 



£/^=Z(^_/.) (12) 



174' THEORY OF FLEXURE. [Art. 24. 

«'' = -r£7 ('4) 

M,= Wl (15) 

The general expressions for the shear and the intensity of 
loading are : 

■5 = ^/gr= W-^px (16) 

^^^ir=/ (17) 



Case II. 

This case, shown in the figure, is that of a non-continuous 
beam, supported at each end, and carrying both a uniform load 



-^ X H 



'J/ 






w 




Fig.2 



(whose intensity is /) and a single weight W at its middle 
point. The reaction R, at either end, will then be : 



Art. 24.] THE NEUTRAL CURVE. 175 

2 

The general value of the moment will then be : 

M=EI^ = Rx-^ .... (18) 



The origin of x and w is taken at A. 
Remembering that : 



dw . I 

— — = o for X ^=^ — . 
dx 2 



and integrating between the limits x and — : 
Again integrating : 



\ \R (x^ xl'\ p fx^ xl\ ) , , 

w^-^x-( — ;-U4 ~"^)(' ■• ^"""^ 



• EI\2 \3 4 



The greatest deflection Wj occurs at the centre of the span, 
for which : 



/ 

X ^^ — 
2 



Hence : 



-■ = -ifi7{^+l^^[ • • • • (^^^ 



17^ THEORY OF FLEXURE. [Art. 24. 



The greatest moment, also, is found by putting : 

/ 



X ^^ — 

2 



It has the value : 



^•=f (^+ft ^''> 



These formulae are made applicable to a non-continuous 
beam carrying a uniform load only, by putting W = o. They 
then become : 



_ /^ 



R = 

2 



M^El'^'^^i^il-.) (.3) 



dx 



Pj-dw _ p fxH x^ I'' 



dx 2\2 x 12/**'*^'^ 



^ "^ ^dm ^^-^'^ - ^ - M .... (25) 



■^. = ^' ....".. (27) 

The formulae for a beam of the same kind carrying a single 
weight at the centre, are obtained by putting/ = o in the first 



Art. 24.] THE NEUTRAL CURVE. 177 

set of equations. Those for the greatest deflection and great- 
est moment, only, however, will be given. They are : 

Wl 

^. = -^ (29) 

4 

The general values of the shear and intensity of loading 
are : 



Case III. 

The general treatment of continuous beams requires the 
use of the theorem of three moments. The particular case to 



c 



Fig. 3 



be treated is shown in Fig. 3. The beam covers the three 
spans, DA^ AB and BC^ and is continuous over the two points 
of support A and B, 
12 



178 THEORY OF FLEXURE. [Art. 24. 



Let DA = l,^ 

" AB = L 



'' BC = l,, 



Let 4 = nl^ = 71 1 y 



Let the intensity of the uniform load on AB be represented 
by/ and let the two single forces P and P' only, act in the 
spans DA and BC respectively. Also let the two distances : 



DE = 2^ = al^ and CF = a I, 



be given. It is required to find the magnitudes of the forces P 
and P\ if the beam is horizontal at A and B. 

Since the beam is horizontal at A and B, the bending mo- 
ments over those two points of support will be equal to each 
other, for the load on AB is both uniform and symmetrical. 
Let this bending moment, common to A and B, be represented 
by M^. As the ends of the beam simply rest at D and Cy the 
moments at those two points reduce to zero. 

Because the four points D, A, B and C are in the same 
level, the first member of Eq. (14), of Art. 23, becomes equal 
to zero. 

If that equation be applied to the three points D, A and B, 
the conditions of the present problem produce the following 
results : 

Ma = o, AT, = M, = M, 
and 

-lip(//- „~>=/^. 

tc 4 

Hence the equation itself will become : 



Art. 24.] THE NEUTRAL CURVE. 179 

Mi2l, + 34) + -f (4^ - ^.O^x + / V- = o . (32) 
/i 4 

•• "'^ 4/x(2/, + 34) 

1—2 M 

:. Reaction at D = R, = P ' ^ + Ip . . . . (34) 



As the origin of z^ is at D, x will be measured from the 
same point. 

Separate expressions for moments must be obtained for the 
two portions, DE and EA, of Z^, because the law of loading in 
that span is not continuous. 

Taking moments about any point of EA : 

El'^ = R,x-P{x-z^ (35) 

Remembering that : 

dw _ 
dx 

for X = /j, and integrating between the limits x and /, : 

^^ S^ = ^ (-^'^ - ^-'^ - T (^' - ^-'^ + ^^' ^-^ " ^') (36) 

Again, remembering that w = o for x = /j, and integrating 
between the limits x and /j : 



l80 THEORY OF FLEXURE. [Art. 24. 

+>.. (I - /.. + ^) (37) 



Taking moments about any point in DE : 



^^S^ = ^'- (38) 



••• ^^S = ^-f + ^ (39) 



Making x ■= z^ in Eqs. (36) and (39), then subtracting 



c = - 4= A' - I- (^.' - /.O + -P^xfe - A) . 



•■• ^^ S = f (^ " ^-') ~ T (•"■' ~ ''^ + -^^- (^- ~ ^-^ (40) 



Remembering that w = o for ;ir = o, and integrating be- 
tween the limits x and o : 



EIW = I? (^ - /.=;.) - ^ (V - l^)x + P^, (^. - />. (41) 



Making x = z^m Eqs. (37) and (41), then subtracting : 



Rl^ P Pz 

^-^ (/.3 _ ^,3) + ^. (/,. _ ^,>) = O . . (42) 



Art. 24.] 



THE NEUTRAL CURVE. 



181 



Putting the value of M^ from Eq. (33) in Eq. (11), then in- 
serting the value of R^^ thus obtained, in Eq. (42), after 
making z^ = al^ : 



P 



\^ - a) -'^^ -(, - a^)^\ai,^ a^)^^ 



_ pnH^ 



4(2 + zn) ' 



P 



pnH^ 



pnl^ 



6a{i — a^) 6a{i — a"") 



■ ■ (43) 



This is the desired value of P, which will cause the beam to 
be horizontal over the two points of support A and B when 
the span AB carries a uniform load of the intensity/. 

By the aid of Eq. (43), Eq. (33) now gives : 



M,= - pn 



12(2 + 3;^ 



pn'l^^ _ 



12 



-ML . ,„. 



It is to be noticed that M^ is entirely independent of / 
or ly Eq. (43) also gives : 






(45) 



Hence : 



2 ^ 



(46) 



Thus any of the preceding equations may be expressed in 
terms of/ or P, 



1 82 



THEORY OF FLEXURE. 



[Art. 24. 



i?i, also, becomes : 



i?. = 



6a{i -\- a) 12 



(47) 



or : 



7?, = P(i - ^) 



I ail -\- a) 

2 ^ ■' 



. . . (48) 



It is clear that there cannot be a point of no bending in 
DE. Hence, the point of contra-flexure must lie between E 
and A, Fig. 3. In order to locate this point, according to the 
principles already established, the second member of Eq. (35) 
must be put equal to zero. Doing so and solving for x : 



X = 



P 



P-R. 



z. 



(49) 



Since R is always greater than R^^ there will always be a 
point of contra-flexure. 

All these equations will be made applicable to the span BC^ 
by simply writing a' for a^ 4 for /j, and n' for n. 

As an example, let : 



« = — and n = I, 

2 



Eqs. (43), (44) and (47) then give : 



P=l,/; 



M,= -l'l = 



12 



?>PI , 

16 ' 



Art. 24.] THE NEUTRAL CURVE. 1 83 



^■=^^-(|-i) = r6^^=f6^^ 



after writing : 



/, = 4 = 4 = /. 



In general, the span l^ is called '' a beam fixed at one end, 
simply supported at the other and loaded at any point with 
the single weight, P." 

Let it, again, be required to find an intensity, "/'," of a uni- 
form, load., resting on the spa?i /j, which will cause the beam to be 
horizontal at the points A and B. 

Since the load is continuous, only one set of equations will 
be required for the span. The equation of moments will be : 

El'^^^Kx-^'^ (50) 

Integrating between the limits x and l^ : 

ir/g = f-(.-/.0-{(.-/.3). . . (5.) 

Integrating between the limits ;t* and o : 
But, also, w^= o when x = l^. Hence : 

^-3=8 •■• ^-^l^'^- • • • (53) 



1 84 THEORY OF FLEXURE, [Art. 24. 

This equation gives the value R^ when/' is known. Making 
X = l^\Xi Eq. (50) and using the value of Rj_ from Eq. (53) : 

^^=^"■^(1-0 = --8- • • • • (54) 
Adapting Eq. (32) to the present case : 



M, (2/, + 34) + i (/' 43 + pl^) = o. 

Equating these two values of M^ : 

P' = ^-Pn' (56) 

Thus is found the desired value of /'. In this, case the 
span /j is called "■ a beam fixed at one end, simply supported 
at the other and uniformly loaded." 

The points of contra-flexure are found by putting the 
second member of Eq. (50) equal to zero and solving for ^, 
after introducing the value of R^ from Eq. (47). Hence : 

- l,x — x^ = o, 
4 



or : 



X = o and x = —L 

4 

Between the simply supported end and point of contra- 



Art. 24.] THE NEUTRAL CURVE. 1 85 

flexure the beam is evidently convex downward^ and convex 
upward in the other portion of the spans 4 and l^ whether the 
load is single or continuous. Moments of different signs will, 
then, be found in these two portions, and there will be a maxi- 
mum for each sign. The location of the sections in which 
these greatest moments act may be made in the ordinary man- 
ner by the use of the differential calculus ; but the negative 
maximum is evidently M^, given by Eqs. (44) and (55). On the 
other hand th^ positive maximum is clearly found at the point 
of application of P in the case of a single load, and at the 
point 

^ - g ^n 

in the case of a continuous load. These conclusions will at 
once be evident if it be remembered that the portion of the 
beam between the supported end and point of contra-flexure 
is, in reality, a beam simply supported at each end. These mo- 
ments will have the values : 



^ ^ 4(2 + yt) 



M; = ^^p-k^ . (58) 



In case of a single load if P is given, and not /, Eq. (45) 
shows : 



I — — a(i + a) 

2 ' 



M, = Pi, (i - a)a 
The points of greatest deflection are found by putting the 



1 86 THEORY OF FLEXURE. [Art. 24. 

second members of Eqs. (36), (40) and (48) each equal to zero, 
and then solving for x. They are not points of great impor- 
tance, and the solutions will not be made. 

The following are the general values of the shears for a 
single load on 4 : 

In AE- S = EI~^ = R,- P-, [from Eq. (35)]. 



In ED; S, = El"^^ = R,] [from Eq. (38)]. 



The shear in /^ for the uniform load/' is : 

S' = EI-~ = R,-p'x- [from Eq. (50)]. 



Also : 



Intensity of load = EI — — = — /'. 



As has already been observed, all the equations relating to 
the span l^ may be made applicable to the span ^ by changing 
a to a and n to r^ . 

The span 4 remains to be considered. 

Since the bending moments at A and B are equal to each 
other, and since the loading is uniformly continuous, half of it 
(the load/4) will be supported at A and the other half at B. 
In other words, the vertical shear at an indefinitely short dis- 
tance to the right of A, also to the left of B, will be equal to 

/4 

--— . Let X be measured to the right and from A, The bend- 
ing moment at any section x will be : 



Art. 24.] THE NEUTRAL CURVE. 1 87 

dx"- ' ^ 2 2 



or : 



^^S =M,'r{{l^-x') .... (59) 



Integrating between the limits x and o : 



^^f = ^^^^- + l(-f-T)- • • (^°) 



Again integrating between the same limits : 



^/. = fe+f^(/.3_^) (60 



Since : 



dw _ 
dx 



for 4, Eq. (60) will give 7J/2 independently of preceding equa- 
tions. Following this method, Jtherefore : 



12 



This is the same value which has already been obtained. 
Introducing the value of M^ : 



1 88 THEORY OF FLEXUER. [Art. 24. 

Ej dw ^ ^d^_x2__li\^ . . (63) 
dx 2 V 2 % 6 / ^ ^^ 



Elw =^7/^-|-|-). . . . (64) 



The points of contra-flexure are found by putting the 
second member of Eq. (62) equal to zero. Hence : 





X = iA- ± 



1 + /i _ L \ ) 0.789/,. 

^ ' ' 0.2114. 



The moment at the centre of the span is found by putting, 

4 



in Eq. (62) : 



X = 

2 



24 



This is the greatest positive moment. 
The general value of the shear is : 

dx^ ^ \ 2 / 

And the intensity of load : 



Art. 24.] THE NEUTRAL CURVE. 1 89 

The span 4 is generally called ** a beam fixed at both ends 
and uniformly loaded." 

It is sometimes convenient to consider a single load at the 
centre of the span 4 while the beam remains horizontal at A 
and B ; in other words, to consider '■' a beam fixed at each end 
and supporting a weight at the centre." 

Let W^ represent this weight : then a half of it will be the 
shear at an indefinitely short distance to the right of A and 
left of B. As before, let x be measured from A^ and positive 
to the right. The moment at any point will be : 

Integrating between x and o : 

dw Wx^ 

EI ^ = M^ ---.,.. , {66) 

If4r = ^, then will 
2 

dw 



hence : 



W^4 



8 



The general value of the moment then becomes : 

^=£/^=-g^-— .... (67) 



190 THEORY OF FLEXURE. [Art. 25. 

If ;r = — in this equation, the bending moment at the centre 
(where Wis applied) has the value : 

Wl 

Centre moment = —^ . 

8 

Hence, t/ie bending moments at the centre and ends are each 
equal to the product of the load by one eighth the span^ but have 
opposite signs. 

A second integration between x and o gives : 

I (Wkx^ Wx^\ .._. 

Hence, the deflection at the centre has the value : 

Wl? 



Centre deflection 



ig2EI ' 



By placing M = o, the points of contra-flexure are found at 
the distance from each end : 



4 



Art. 25. — The Flexure of Long Columns. 

A " long column " is a piece of material whose length is a 
number of times its breadth or width, and which is subjected 
to a compressive force exerted in the direction of its length. 
Such a piece of material will not be strained, or compressed, 
directly back into itself, but will yield laterally, as a whole, 
^thus causing flexure. If the length of a long column is many 



Art. 25.] 



LONG COLUMNS. 



191 



times the width or breadth, the failure in consequence of flex- 
ure will take place while the pure compression is very small. 

As with beams, so with columns, the ends may be " fixed," 
so that the end surfaces do not change their position however 
great the compression or flexure. Such a column is frequently, 
perhaps usually, said to have *' flat " ends. If the ends of the 
column are free to turn in any direction, being simply sup- 
ported, as flexure takes place, the column is said to have 
" round " ends. It is clear that if the column has freedom in 
one or several directions, only, it will be a "■ round " end col- 
umn in that one direction, or those several directions, only. It 
is also evident that a column may have one end "" round " and 
one end '^ flat " or "fixed." 

In Fig. I let there be represented a column with flat ends, 
vertical and originally straight. After extcnal pressure is 

armmimm 



imposed at A, the column will take a shape similar 
to that represented. Consequently the load P, at 
A^ will act with a lever arm at any section equal 
to the deflection of that section from its original 
position. Let y be the general value of that de- 
flection, and at B let y^y\. Let x be measured 
from Aj as an origin, along the original axis of 
the column. In accordance with principles already 
established, the condition of fixedness at each of 
the ends A and C is secured by the application of 
a negative moment — M, Now it is known from 
the general condition of the column that the curve 
of its axis will be convex toward the axis of x at 
and near A^ while it will be concave at and near B 
(the middle point of the column). Hence, since j/ 
is positive toward the left, and since the ordinate and its second 
derivative must have the same sign when the curve is convex 
toward the axis of the abscissas, the general equation of mo- 
ments must be written as follows : 



Fig.l 



192 THEORY OF FLEXURE. [Art. 2;. 
- EI ^^ = - M + Py (I) 

Multiplying by — 2dy : 

EI ^^yj^y = 2Mdy - P2y dy. 



.-. EI (J)' = 2My - Py + (. = o) . . . (2) 
^ = o because the column has flat ends, and, 







dy 
dx 




when y = 0. 


Also : 


dy 




when y = y^. 












2 


• •09 



• • (3) 
Eq. (2) now becomes : 



= dx. 



'IE 21/ 

.*. ^ = /^ / -p- ver sin"'' -^ . ... (4) 



liy=y, : 



Art. 25.] LONG COLUMNS. 193 

/ [ET , , 



In this equation / is the length of the column. From Eq. 
(5) there may be deduced : 

P=^^ (6) 

It is to be observed that Pis wholly independent of the de- 
flection^ i. e.y it remains the same, whatever may be the amount 
of deflection, after the column begins to bend. Consequently, 
if the elasticity of the material were perfect, the weight P 
would hold the column in any position in which it might be 
placed, after bending begins. 

Eq. (6) forms the basis of *' Hodgkinson's Formula " for 
the resistance of long columns, of which more will be given 
hereafter. It was first established by Euler. 

Some very important results flow from the consideration of 
Fig. I in connection with the preceding equations. 

The bending moment at the centre, B, of the column is ob- 
tained by placing J =/i in Eq. (i) ; its value is, consequently : 

M' = - M -^ Py, = M (7) 

Hence the bending at the centre of the column is exactly the 
same {but of opposite sign) as that at either end. Between A 
and B, then, there must be a point of contra-flexure. 

Putting the second member of Eq. (i) equal to zero, and 
introducing the value of M from Eq. (3) : 

13 



194 THEORY OF FLEXURE. [Art. 25. 

Introducing this value of j in Eq. (4), and bearing in mind 
E* (5) : 



-f\/¥ = T » 

The points of contra-flexure, then, are at H and D, — I and 

^ / from A. 
4 

Hence, the middle half of the column {HD) is actually a 

column with round ends, and it is equal in resistance to a fixed- 
end column of double its length. 

Hence writing /' for - and putting 2I' for /in Eq. (6) : 



P=~jT-- ....... (9) 



Eq. (9) gives the value of P for a round-end column. 

Again, either the upper three quarters {AD) or the lower 
three quarters {CH) of the column is very nearly equivalent to 
a column with one end flat and one end round, and its resist- 
ance is equal to that of a fixed-end column whose length is — 
its own. Putting, therefore : 



and introducing : 



in Eq. (6) : 



/. = ^/ 



/ =4/, 

3 



Art. 25.] LONG COLUMNS. I95 

P=2.25-— - (10) 

''X 

The last case is not quite accurate, because the ends of the 
columns HC and AD are not exactly in a vertical line. 

In reality, the column under compression may be composed 
of any number of such parts as HD, with the portions //J/ and 
CD at the ends, thus taking a serpentine shape, so far as pure 
equilibrium is concerned. In such a condition the column 
would be subjected to considerably less bending than in that 
shown in the figure. In ordinary experience, however, the 
serpentine shape is impossible, because the slightest jar or 
tremor would cause the column to take the shape shown in 
Fig. I. Hence, the latter case only has been considered. 

If r is the radius of gyration and S the area of normal sec- 
tion of the column, Eqs. (6) and (9) will take the forms : 



— and — 



5 P S /' 

Eq. (10) will, of course, take a corresponding form. 

P 

These equations evidently become inapplicable when -^ 

approaches C, the ultimate compressive resistance of the ma- 
terial in short blocks. The corresponding values of (-) at 
the limit, are : 



(■') 



for fixed and round ends respectively ; other conditions of 
ends will be included between those two. 




196 THEORY OF FLEXURE. [Art. 26. 

If, for wrought iron : 

E = 28,000,000 and C = 60,000, 

the above values become 136 and 68, nearly. 

Euler's formula, therefore, is strictly applicable only to 
wrought-iron columns, with ends fixed or rounded, for which 
/ -^ r exceeds 136 and 68, respectively. 

If, for cast iron : 

E = 14,000,000 and C = 100,000, 
Eqs. (i i) give : 

— = 74, and — = 37, nearly. 

Euler's formula evidently becomes Inapplicable consider- 
ably above the limits indicated, since columns in which — has 

r 

those values will not nearly sustain the intensity C. 

The analytical basis of " Gordon's Formula " for the re- 
sistance of long columns is so closely associated with the 
empirical, that both will be treated together, hereafter. 

Art. 26. — Graphical Determination of the Resistance of a Beam. 

The graphical method is well adapted to the treatment of 
beams whose normal sections are limited either wholly or in 
part by irregular curves. In Fig. i is represented the normal 
section of such a beam, the centre of gravity of the section 
being situated at C. The lines HL, AB and DF are parallel. 
As is known by the common theory of flexure, the neutral axis 
will pass through C. 



Art. 26.] 



GRAPHICAL METHOD. 



197 



Let aa be any line on either side oi AB, then draw the lines 
aa' normal to AB, having made MN and UK equidistant from 
AB. From the points a, thus determined, draw straight lines 
to C. These last lines will include intercepts, di?y on the orig- 
inal lines aa. Let every linear element parallel to AB, on each 




side of Cy be similarly treated. All the intercepts found in this 
manner will compose the shaded figure. 

This operation, in reality, and only, determines an amount 
of stress with a uniform intensity identical with that developed 
in the layer of fibres farthest from the neutral axis, and equal 
to the total bending stress existing in the section ; this latter 
stress, of course, having a variable intensity. NL represents 
the layer of fibres farthest from the neutral surface, conse- 
quently MNwas taken at the same distance from AB. Any 
other distance might have been taken, but the intensity of the 
uniform stress would then have had a value equal to that 
which exists at that distance from the neutral axis. Again, a 



198 THEORY OF FLEXURE. [Art. 25. 

different intensity might have been chosen for the stress on 
each side of AB. It is most convenient, however, to use the 
greatest intensity in the section for the stress on both sides of 
the neutral axis ; this intensity, which is the modulus of rupt- 
ure by bending, will be represented, as heretofore, by K. 

Let c and c be the centres of gravity of the two shaded 
figures. These centres can readily and accurately be found by 
cutting the figures out of stiff manilla paper and then balanc- 
ing on a knife edge. Let s represent the area of the shaded 
surface below AB^ and s the area of that above AB. 

Because this is a case of pure bending, the stresses of ten- 
sion must be equal to those of compression. Hence : 

Ks = Ks' \ or, s ■= s' (i) 

The moment of the compression stresses about AB will 
be : 

Ks X cC. 

The moment of the tensile stresses about the same line 
will be : 

Ks X cC, 

Consequently the resisting moment of the whole section 
will be : 

M = Ks {c'C + cC) = Ks X CC . . . . (2) 

Thus, the total resisting moment is completely determined. 
In some cases of irregular section the method becomes ab- 
solutely necessary. 

It is to be observed that the centre of gravity, c or c\ is at 



Art. 27.] UNEQUAL COEFFICIENTS OF ELASTICITY. I99 

the same normal distance from AB as the centre of the actual 
stress on the same side of AB with c or c'. 



Art. 27. — The Common Theory of Flexure with Unequal Values of 

Coefficients of Elasticity. 

In all cases it has hitherto been assumed that the coeffi- 
cient of elasticity for tension is equal to the same quantity for 
compression. In reality, this is exactly true for probably no 
material whatever, though* the error, fortunately, is not serious 
for the greater portion of the material used by the engineer. 
By the aid of the assumptions used in the common theory of 
flexure, formula involving this difference of coefficients may 
be deduced. As these are of little real value, however, a few 
general results, only, will be obtained. 

Let £ represent the coefficient of elasticity for tension. 

Let E' represent the coefficient of elasticity for compres- 
sion. 

As has before been assumed, the normal sections of the 
beam, which are plane before flexure, will be taken as plane 
and normal to the neutral surface after flexure. Also, as be- 
fore (Art. 18), let II represent the rate of strain (strain for unit 
of length of fibre) at unit's distance from the neutral surface ; 
let the variable width of the section be represented by i?, while 
/ represents the variable normal distance of the element ddy 
from the neutral axis of the section. The element of the ten- 
sile stress in the section will be : 

£nf . d dy. 

The elementary moment of the same will be : 

Euy"^ b dy. 



200 



THEORY OF FLEXURE. 



[Art. 27. 



In precisely the same manner, the elementary compressive 
moment will be : 

E 'uy^b dy. 

Consequently, the total resisting moment will have the 
value : 



M =^ u 



Pi r° 

E y^b dy -\- E' y^b dy 

L_ Jo •'-^I 






h K'r 

y^b dy + — - y^b dy 

y J-v 



(0 



The ordinates y and y are those belonging to the extreme 
fibres of the section, while iT and K' represent stress intensi- 
ties in those fibres. The general value of y is also affected 
with the negative sign on the compression side of the beam. 

It has been shown in Art. 18 that : 



IL = 



also, in the case of straight beams, that : 



I 
9 



d^w 
dx'^ 



w being the deflection and x the abscissa measured along the 
axis of the beam. For the sake of brevity, let the quantity in 
the brackets in the second member of Eq. (i) be represented 
by ^y, in which, consequently, E' will be displaced by nE, n 
being the ratio between E and E^. Eq. (i) may then take the 
form : 



Art. 27.] UNEQUAL COEFFICIENTS OF ELASTICITY. 201 

or : 



-^y dx' 



(3) 



If Mand/sire expressed in terms of x, w may at once be 
found. If, as is usual, the section is uniform, then will / be 
constant and M, only, will be a function of x. 

If the section is rectangular, d will be constant andy will 
take the following value : 

3 3 



r 

f=- b\' + ny^) (4) 



Because the internal tensile stress in any section must equal 
the internal compressive stress in the same section : 



E21 



ry' 
dj/ dj^ = £ 'ic ' by dy (5) 



Eq. (5) will enable the neutral axis of any section to be 
located. If the section is symmetrical, the neutral axis will 
evidently be situated on that side of the centre of gravity of 
the section on which is found the greatest coefficient of elas- 
ticity. 



202 THEORY OF FLEXURE. [Art. 28. 

Art. 28. — Greatest Stresses at any Point in a Beam. 

If the approximate conditions on which are based the 
formulae found in the latter part of Art. 17 are assumed, some 
interesting and important results may be very easily obtained. 

The Eqs. (13), (14) and (15) of Art. 6 are those which lead 
to the ellipsoid of stress, and hence to all of its special cases 
and consequences. The equation representing the ellipsoid of 
stress might first be found, and then the special form relating 
to the case considered. It will be more simple and direct, 
however, to use those equations immediately. 

If, as in Art. 17, a rectangular beam carrying a load at its 
end be assumed, in which : 

7; = 7; = A^, = iV3 = o, 

Eqs. (13), (14) and (15) of Art. 6 reduce to : 

N^ cos p -\- T2 cos r ^=^ P cos n ; 

T2 cos p = P COS p. 

But since all stress is. assumed to be found in planes paral- 
lel to ZX : 

cos r = sin p, and cos p = sin tt. 

Hence : 

Wj cos p + T2 sin p = P cos 7t (i) 

7*2 cos p = P sin 7t (2) 

in which P is the intensity of the resultant stress on any plane 



Art. 28.] GREA TEST STRESS A T ANY POINT. 203 

at any point ; / the angle which the normal to that plane 
makes with the axis of X (the axis of the beam) ; and n the 
angle which the direction of P makes with the same axis. 

Let it first be required to find the plane, at any point, on 
which the 7iornial or direct stress is the greatest. 

It is known from the theory of internal stress that this 
greatest normal stress will be the resultant and, hence, a prin- 
cipal stress. Hence the relation : n =^ p \ or: 

iV, + T,tanp = P (3) 

7; ^Ptanp (4) 

If i^ is the weight carried by the beam at its end; /the 
moment of inertia of the beam's cross section ; and d its half 
depth, or greatest value of z, it has been shown in Arts. 17 
and 18 that : 

N, = ^,, and T,^^(d'-^) ... (5) 

Inserting the value of P from Eq. (4) in Eq. (3) : 
7*2 — T^ tan"" p = N^ ta7t p. 



N 
,\ tan^ p + -^ tanp = i. 

-^ 2 



Solving this quadratic equation and then inserting the 
values of T^ and N^ from Eq. (5) : 



tan /> = _^ 

^ d^ — z" d^ — z" 



204 THEORY OF FLEXURE. [Art. 28. 

This value of tan p put in Eq. (3), or Eq. (4), will give the 
greatest value of the direct or normal stress (also resultant) at 
any point in the beam. 

At the exterior surface, d =^ z ] hence : 

tanp = o or -co. 

Since for this point T^ = o, the first value gives, by Eq. (3), 
P= iVj. The second value, by Eq. (4), gives, P= o. These 
results might have been anticipated. 

At the neutral surface, ^ == o; hence : 

tan p r= ± I = tan ± 45°. 

Hence, at the neutral surf ace there are two planes on whicJi the 
stress is wholly normal, and these planes make angles of 4.^° with 
the 7ieutral surface, or 90° with each other (i.e., tJiey arc priii- 
cipal planes). 

Since iVj = o at the neutral surface, either of the Eqs. (3) 
or (4) gives : 

Fd"" 
P=±T^=±-~ (7) 



Hence each of these normal or principal stresses equals in inten- 
sity that of the trafisverse or longitudi7ial shear at the neutral 
surface ; also, one of these principal stresses is a tension and the 
other a compression. 

2/ [xz ^ ^{d^ - zj + x'z' S 
is the equation of the locus of the point of constant greatest 



Art. 28.] 



GREA TEST STRESS A T ANY POINT. 



205 



normal intensity of stress, if P be taken constant and equal to 
any possible value. 

Let it next be required to find the plane of greatest shear at 
any point in the beam, and the value of that shear. 

The shear on any plane will be : 



P sin {^Tt — p) ^=1 T 



(8) 



Multiplying Eq. (i) by (— si7i p) and Eq. (2) by cos p, then 
adding : 

— N^ cos p sin p -j- T7. (cos^ p — sin^ p) = P{sin it cos p 
— cos 7t sin p) =: P sin (jt — p) ^ T. 



AT 
T = — — ^ sin 2p + T^ cos 2p . ... (9) 



It is now required to find what value of / will make the 
general value of T [given by Eq. (9)] a maximum. Hence : 



dT 
dp 



—-. — = — iVj cos 2/ — 2 7^2 sin 2p ■=■ o. 



tan 2p ■= — 
.*. cos 2/ = ± 



2T. 



xz 



d' - z^ 



r . . . (10) 



Vx^z^ + {d"" - zj 



Eqs. (10) give the value of/ which is to be placed in Eq. 
(9), in order to obtain the greatest value of T at any point of 
the beam. 



206 THEORY OF FLEXURE. [Art. 28. 

From Eq. (9) : 

( N ) 

T — T^ cos 2p \ — ■ — —■ tan 2p -\- \\ 

.*. T= ± ^ Vx'^-\-{d--2j .... (11) 

At the exterior surfaces of the beam : 

2 — ± d. 
Hence : 

2/ 2 

For this case, also : 

cos 2/ = o, or / = 45°. 

Hence, at the exterior surfaces of the beam the planes of 
greatest shear make angles of 45° with the axis of the beam^ 
and the intensity of the shear is half that of the direct stress at 
the same place. 

At the neutral surface : ^ = o. Hence : 



Fd"" 
T = ± — Y == Zj ; and cos 2p = ± i. 



Hence, 2/ = o or 180° ; or p = o or 90° ; i.e., the planes of 
greatest shear are the transverse and longitudinal planes, and 
the greatest shear itself is, consequently, the transverse or lon- 
gitudinal shear. 



Art. 28.] GREATEST STRESS AT ANY POINT. 20/ 

If T is given any possible value and considered constant, 
Eq. (i i) will give the locus of the point of constant greatest 
shear. 

The result expressed in Eq. (7) Is of great value in deter- 
mining the thickness of the web of flanged beams, as will be 
seen hereafter. 



PART II.— TECHNICAL. 



CHAPTER V. 
Tension. 



Art, 29. — General Observations. — Limit of Elasticity. 

Hitherto, certain conditions affecting the nature of elastic 
bodies and the mode of applying external forces to them, have 
been assumed as the basis of mathematical operations, and 
from these last have been deduced the formulae to be adapted 
to the use of the engineer. These conditions, in nature, are 
never realized, but they are approached so closely, that, by the 
introduction of empirical quantities, the formulae give results 
of suf^cient accuracy for all engineering purposes ; at any rate, 
they are the only ones available in the study of the resistance 
of materials. 

In determining the quantity called the '* coefificient of elas- 
ticity," it is supposed that the body is perfectly elastic, Le.^ that 
it will return to its original form and volume when relieved of 
the action of external forces, also, that this " coefficient " is 
constant. There is reason to believe that no body known to 
the engineer is either perfectly elastic or, possesses a perfectly 
constant coefficient of elasticity. Yet, within certain not well 
defined limits the deviations from these assumptions are not 
sufficiently great to vitiate their ^r^dX practical usefulness. 



Art. 29.] LIMIT OF ELASTICITY. 209 

The " not well defined " limit for any one given material is 
called its " limit of elasticity," or '' elastic limit." The '' limit 
of elasticity," then, may be defined as that degree of stress 
within which the coefficient of elasticity is essentially constant 
and equal to the stress divided by the strain. 

In some materi-als, like many grades of wrought iron and 
steel, the limit of elasticity approximates, to a greater or less 
degree, to the condition of a well defined point. If a piece of 
such a material is subjected to stress in a testing machine, at 
the elastic limit, the amount of strain caused by a given incre- 
ment of stress will be observed, comparatively speaking, to 
rapidly increase. This increase may be uniform for a consider- 
able range of stress, but it finally becomes irregular, after 
which failure takes place. 

In other materials, there seems to be no simple relation 
between stress and strain for any condition of stress whatever. 
For such a material it obviously is impossible to assign either 
any definite elastic limit or coefficient of elasticity. 

Between these limits, of course, all grades of material are 
found. 

It should be stated that some authorities have given arbi- 
trary definitions of the elastic limit, and that these definitions 
have been very much used. Wertheim and others have con- 
sidered the elastic limit to be that force which produces a per- 
manent elongation of 0.00005 of the length of a bar. Again, 
Styffe defines, as the limit of elasticity, a much more compli- 
cated quantity. He considers the external load to be gradu- 
ally increased by increments, which may be constant, and that 
each load, thus attained, is allowed to act during a number of 
minutes given by taking 100 times the quotient of the incre- 
ment divided by the load. Then the " limit of elasticity " is 
" that load by which, when it has been operating by successive 
small increments as above described, there is produced an in- 
crease in the permanent elongation which bears a ratio to the 
length of the bar equal to o.oi (or approximates most nearly 



210 TENSION. [Art. 30. 

to o.oi) of the ratio which the increment of weight bears to 
the total load." (Iron and Steel, p. 30.) 

The most natural value, however, seems to be that stress 
which exists at the point where the ratio between stress and 
strain ceases to be essentially constant, though the assignment 
of the precise point be difficult in many cases and impossible 
in some ; and in that sense it is here used, though seldom in 
ordinary testing. 

Again, in the common theory of flexure, modes of appli- 
cation of external forces and a constitution of material are 
assumed, which are never realized ; yet the resulting formulae 
are of inestimable value to the engineer. 

Finally, it will be shown in the first section of Art. 32 that 
it is in general impossible to produce a uniform intensity of 
stress in a normal cross section of a body subjected to pure 
tension, and, consequently, that the ultimate resistance, as 
experimentally determined, is a mean intensity which may be, 
and usually is, considerably less than the maximum sustained 
by the test piece. 

These general observations are to be carefully borne in 
mind in connection with all that follows. 



Art. 30. — Ultimate Resistance. 

After a piece of material, subjected to stress, has passed its 
elastic limit, the strains increase until failure takes place. If 
the piece is subjected to tensile stress, there will be some de- 
gree of strain, either at the instant of rupture or somewhat 
before, accompanied by an intensity of stress greater than that 
existing In the piece in any other condition. This greatest in- 
tensity of internal resistance Is called the '* Ultimate Resist- 
ance." 

In very ductile materials this point of greatest resistance is 
found considerably before rupture ; the strains beyond it in- 



Art. 31.] DUCTILITY AND SET. 211 

creasing very rapidly while the resistance decreases until sepa- 
ration takes place. 

The ultimate resistances of different materials used in en- 
gineering constructions can only be determined by actual tests, 
and have been the objects of many experiments. 

It has been observed in these experiments that many in- 
fluences affect the ultimate resistance of any given material, 
such as mode of manufacture, condition (annealed or unan- 
nealed, etc.), size of normal cross section, form of normal cross 
section, relative dimensions of test piece, shape of test piece, 
etc. In making new experiments or drawing deductions from 
those already made, these and similar circumstances should all 
be carefully considered. 



Art. 3i. — Ductility. — Permanent Set. 

One of the most important and valuable characteristics of 
any solid material is its ** ductility," or that property by which 
it is enabled to change its form, beyond the limit of elasticity, 
before failure takes place. It is measured by the permanent 
" set," or stretch, in the case of a tensile stress, which the test 
piece possesses after fracture ; also, by the decrease of cross 
section which the piece suffers at the place of fracture. 

In general terms, i.e.^ for any degree of strain at which it 
occurs, " permanent set " is the strain which remains in the 
piece when the external forces cease their action. It will be 
seen hereafter that in many cases, and perhaps all, permanent 
set decreases during a period of time immediately subsequent 
to the removal of stress. Indeed, in some cases of small strains 
it is observed to disappear entirely. 

Some experimenters, with the aid of very delicate meas- 
uring apparatus, have observed permanent set even within what 
is ordinarily termed the limit of elasticity, and have been led 
to believe that a very small permanent set exists with any de- 



212 WROUGHT IRON IN TENSION. [Art. 32. 

gree of stress whatever. In such cases, however, it is probable 
that the greater part or all of the permanent set disappears 
after the lapse of a few hours. 



Art. 32. — Wrought Iron. — Coefficient of Elasticity. 

Before considering the experimental results which are to 
follow, it will be interesting as well as important to examine 
some of the circumstances which attend the experimental de- 
termination of the coefficient of elasticity. 

If tensile stress is uniformly distributed over each end of 
a test piece, it will not be so distributed over any other normal 
section. For since lateral contraction takes place, the exterior 
molecules of the piece must move towards the centre. But if 
this motion takes place, the molecules in the vicinity of the 
centre must be drawn farther apart, or suffer greater strains, 
than those near the surface. 

Hence the stress will no longer be uniformly distributed, 
but the greatest intensity will exist at the centre and the least 
at the surface of the piece. These effects will evidently in- 
crease, for a given kind of cross section, with its area. But the 
stretch, or strain, from which the coefficient of elasticity is 
computed, is measured on the surface of the piece, and corre- 
sponds, as has just been shown, to an intensity of stress less 
than the mean, while the latter is actually used in the compu- 
tation. In the notation of Eq. (i), Art. 2,/ is too great and / 
too small ; hence E will be too large. 

As these effects increase with the area of the cross section, 
while other things are the same, larger bars should give greater 
coefficients of elasticity than smaller ones. 

These effects will evidently be intensified, also, if the ex- 
ternal force is applied with its greatest intensity near, or at, 
the centre of the bar, as is the case in testing eye-bars. 

Again, on the other hand, if the ends of the test piece are 



Art. 32.] 



COEFFICIENT OF ELASTICITY. 



213 



gripped on the surface, or skin, as is usually the case with 
small pieces, these effects will be very much modified, and 
possibly entirely counteracted, so that the greatest intensity 
will exist at the surface. In the latter case, the resulting co- 
efficient would be too small. 

Between these extreme cases, all grades will be found. 

From these considerations, it is clear that the manner of 
gripping the test piece, length, character and area of cross sec- 
tion all affect the value of the coefficient of elasticity, and 
should be given in connection with the latter. 

These conclusions apply to any other material, as well as 
to wrought iron. 

Table I. gives the results of some experiments made by the 
Phoenix Iron Co., of Phoenixville, Penn., on some flats and 
rounds of the dimensions shown in the column headed "" Size." 

TABLE I. 



NO. OF BARS. 


SIZE. 


LENGTH. 


STRETCH. 


/• 


E. 




Inches. 


Ft. 


In. 


Inches. 


Pounds. 


Pounds. 


12 


4 X i5 


35 





0.2692 


20,000.00 


31,203,000.00 


9 


4 X ii% 


27 


6 


0.2033 




. 


32,464,700.00 


24 


3f X li 


35 





0.2500 






33,600,000.00 . 


24 


32 X li 


35 





0.2617 




u 


32,098,000.00 


23 


3 X 1 


35 





0.2587 






32,470,000.00 


24 


3 X f 


35 





0.2633 






31,902,000.00 


24 


2x1 


24 


92^ 


0.1948 






30,544,000.00 


36 


2J0 




9 


0.0953 




( 


29,380,000.00 


68 


2^0 




II 


0.0998 




( 


28,056,000.00 


120 


2I0 




9 


0.0947 




( 


29,567,000.00 


48 


2.^0 




9 


0.0955 






29,319,000.00 


72 


2I0 




9 


0.0940 






29,787,000.00 


48 


25O 




9 


0. 1008 


a a 


^i^innn-oo 



The column '^ p " is the intensity per square inch which 
caused the stretches shown in the column headed " Stretch." 
From Eq. .(i) of Art. 2 : 



214 WROUGHT IRON IN TENSION. [Art. 32. 

E = ^ (.) 



In this case, for any individual bar 

Stretch 



/ = 



Length 



"^ 



remembering that the stretch and length must be reduced to 
the same unit. 

Let the above formulae be applied to the twenty-four bars 
3 X ^ inches X (35 ft. = 420 ins.) long. 



^ 20000 X 420 ^ , 

zi = ^ = ^iygo2, 000.00 founds. 



The other values are found in precisely the same way. 
The quantities in the column E are the averages of the num- 
ber of experiments given in the extreme left hand column. 
The fact that the results are the averages of a great number 
of experiments gives the table peculiar value. This table 
is taken from '^ Useful Information for Architects and En- 
gineers/' published by the Phoenix Iron Co. The following 
reference to the table is taken from the same source : *' The 
annexed table gives the results attained in testing with the 
proof load of 20,000 pounds per square inch, a number of bars 
for the International Bridge over the Niagara River, near 
Buffalo, N. Y. The recovery of each bar, after the removal of 
the load, was perfect, no permanent set occurring at less than 
25,000 pounds. It will be observed that the stretch per foot of 
the flat bars is less than that of the rounds, giving them higher 
moduli of elasticity." It is interesting and important to ob- 
serve this last point. 



Art. 32.] 



COEFFICIENT OF ELASTICITY. 



215 



It is to be observed, finally, that these coefficients of elas- 
ticity are determined for one intensity of stress, only, i.e., 
20,000.00 pounds per square inch. It is probable that values a 
little different might be given by other intensities. 

Table II. contains coefficients of elasticity for tension, in 
pounds per square inch, computed from data given in the 
*' Report of the Committee of the Franklin Institute . . . Part 
II., containing the Report of the sub-committee to whom was 
referred the examination of the strength of the materials em- 
ployed in the construction of steam boilers." 

TABLE II. 



< 


LENGTHWISE 














« 




MODE OF 















OR 




L. 


LI. 


/. 


T. 


£. 




MANUFACTURE. 












6 
z 


CROSSWISE. 




















Inches. 


Inches. 


Pounds. 


Pounds, 


Pounds. 


32 


Cross. 


Rolled 


23.00 


.00883 


12,768 


■ 


33,280,000 


32 


" 


" 


23.00 


.0174 


19,152 




25,220,000 


32 


" 


(1 


23.00 


.0274 


25,536 


■ 52,200 


21,450,000 


32 


" 


(i 


23.00 


.039 


31,920 


18,832,000 


32 


" 


'' 


23.00 


.0525 


38,304 




16,777,000 


32 


" 


(( 


23.00 


.0669 


44,688 




15,373.000 


48 


Length. 


Hammered 


24.5 


.01354 


26,600 




48,146,000 


48 


" 


^* 


24.5 


,02882 


39,900 




33,955,000 


48 
48 


" 




24.5 
24.5 


.047 
.072 


50.469 
52,725 


■59,710 


26,294,000 
17,927,000 


48 


" 


it 


24.5 


.0958 


55.100 




14,106,000 


48 


" 


(i 


24.5 


.144 


57.832 




9,831,400 


27 


«t 


Rolled 


24.2 


.0297 


30,840 


55.636 


25,168,000 


51 


Cross. 


Hammered 


24.0 


.0449 


28,250 


59.656 


15,086,000 


53 


" 


^ * 


24.0 


.0314 


26,270 


56,062 


20,070,300 


56 


" 


Puddled 


23-7 


.0059 


28,000 


58,964 


112,476,000 


65 


" • 


" 


24.25 


.0161 


31,540 


51.255 


47,499,000 


85 


Length. 




22,10 


.0127 


25,280 


54,518 


44,161,000 





This report was published in the " Journal of the Franklin 
Institute," for 1837. The data for the computations were taken 
from pages 260 to 275 of Vols. XIX. and XX. of that journal. 
The test specimens were about 0.25 square inch in' area of 
normal section and were cut from boiler plate across or along 
the fibre, as is indicated in the table. The experiments were 



2l6 WROUGHT IRON IN TENSION. [Art. 32. 

made in 1832 and 1833. About 5 per cent was allowed for the 
friction of the machine. It is presumed from the report (which 
is not very clear on some important points) that the smallest 
values of "/," in the various experiments, indicate the elastic 
limit of the different bars. 

The column LI gives the extensions of the bars whose 
lengths appear in the column L, 

The column "/ " gives the intensities of stress per square 
inch for which the corresponding values of the coefficients of 
elasticity (in the column E) have been computed. 

In computing E^ there is to be put in Eq. (i) : 

Bars No. 32 and 48 show the decrease of the coefficient of 
elasticity for degrees of stress up to the ultimate resistances 
per square inch of original section shown in column T. 

The values of E can scarcely be considered more than ap- 
proximate, since the manner of holding the specimen, taken in 
connection with the method of measuring Z/, would probably 
make the elongation either somewhat greater or less than that 
belonging to the length L, This might make some of the 
values of E greater than they ought to be. That belonging to 
bar 56 apparently indicates an error, but it is believed not to 
be in the computations. Undoubtedly, however, many of the 
irons, at that early day, were very stiff and hard! 

Prof. Woodward, in *^ The Saint Louis Bridge," gives the 
results of 6j experiments on specimens varying from 6 to 18 
inches long and from 0.45 inch to 1.13 inches in diameter, from 
17 different producers. In these results the range of variation 
was very great ; in fact the coefficient of tensile elasticity varied 
from 9,500,000 lbs. per sq. in. to 65,500,000, and some of the 
widest variations were in specimens of the same brand. 

Table III. gives the results of the experiments of Mr. Eaton 



Art. 32.] 



COEFFICIENT OF ELASTICITY. 



217 



TABLE III. 



Tensile Experiments on two Annealed '-^Besf^ Wrought Iron Bars 
ten feet long ufid one inch square. 



BAR NO. I. 




BAR 


NO. 2. 




P- 


LI. 


Sets. 


E. 


/. 


LI. 


Sets. 


E. 




Inches. 


Inches. 






Inches. 


Inches. 




2,668 


.00986 




32,457,000 


1,262 


.00520 




29,125,000 


5.335 


,02227 




28,198,000 


2,524 


.01150 




26,347,000 


8,003 


.03407 


.000305 


28,180,000 


3,786 


.01690 


.00050 


26,851,000 


10,670 


•04556 


,000407 


28,101,000 


5,047 


.02240 


.00060 


26,990,000 


13.338 


.05705 


.000509 


28,056,000 


6,309 


.02772 


.00050 


27,312,000 


16,005 


.06854 


.000610 


28,020,000 


7.571 


.03298 


.00045 


27,551,000 


18,673 


.07993 


.000813 


28,033,000 


8.833 


.03790 


.00050 


27,953,000 


21,340 


.09193 


.001525 


27,855,000 


10,095 


.04300 


.00050 


28,198,000 


24,008 


.10485 


.003966 


27,475,000 


11,357 


.04854 




28.077,000 


26,676 


.12163 


.009966 


26,308,000 


12,619 


.05370 


.00070 


28,199,000 


29,343 


.15458 


.031424 


22,782,000 


13,880 


.05950 




27,984,000 


32,011 


.26744 




14,361,000 


15,142 


.06480 




28,041,000 




.28271 


.13566 




16,404 


.06980 




28,186,000 




in 5 minutes 






17,666 


•07530 


,00130 


28,153,000 


34,678 
^ 37,346 
Repeated 


.5148 


.36864 
1.01695 
1.02966 


8,083,000 


18,928 
20,190 


.08170 
.08740 


.00270 


27,794,000 
27,734,000 


1.095 
I. 1949 


4,077,000 


21,452 
22,713 


.09310 
.09920 


.00410 


27,644,000 
27,474,000 


40,013 


1 . 220 
in 5 minutes 


1.093 


3,924,000 


23,795 
25,237 


,10570 
.11250 


.00680 


27,213,000 
26,919,000 


Repeated 


T.411 






26,499 


. 1 2040 




26,420,000 


and left on 


after i hours 






27,761 


.12880 


.0120 


25,872,000 










29,023 


.14500 




23,986,000 




1.424 






30,285 


,1991 




18,244,000 




after 2 hours 






30,285 


.2007 




18,030,000 


i( 


1.433 
after 3 hours 








after 5 min. 












30,285 


.2018 


.0736 




" 


1.434 








after 10 min. 








after 4 hours 






30,285 


.2054 


.0774 




<( 


1.436 








after 15 min. 








after 5 hours 






Repeated 


,2080 


.0796 




(( 


T.437 








after 20 min. 








after 6 hours 






(( 


.2oq6 


.0814 




<( 


1.443 








after i hour 








after 7 hours 






<( 


.2366 


.1082 




C( 


1.443 








after 17 hours 








after 8 hours 






31,546 


.242 


.1083 


15,617,000 


(( 


1.443 








after 5 min. 








after 9 hours 






Repeated 


.2449 


.1111 




(( 


1.443 






• 


after 5 min. 








after xo hours 






32,808 


.5506 


.4141 


7,132,000 


42,681 


2.148 
in 5 minutes 


1.983 


2,384,000 


Repeated 


.7024 
after 5 min. 


.5635 


•» 


Repeated 


. 2.339 
in 5 minutes 






i( 


.70966 
after 10 min. 


.6558 





2l8 



WROUGHT IRON IN TENSION 



[Art. 32. 



TABLE III.— Continued. 



BAR NO. I. 




BAR 


NO, 2. 




P- 


LI. 


Sets. 


E. 


/• 


LI. 


Sets. 


E. 




Inches. 


Inches. 






Inches. 


Inches. 




Repeated 


2.383 
in 10 minutes 


2.212 




Repeated 


1 .014 
after 15 min. 


.366 




it 


2.428 
after 46 hours 


2.237 




34,070 


1.346 
after i min. 




2,839,000 


45,348 


2.580 
after 5 min. 


2.377 


2,109,000 


34.070 


1 .400 
after 2 min. 






Repeated 


2.605 
after i hour 






34,070 


1.600 


1.44 




i( 


2.606 






Repeated 


after 1 mm. 








after 2 hours 






















(( 


1.786 


1.628 






2.606 


2.403 






after i hour 








after 19 hours 














48,016 








35,332 


2.04 


1.874 


2,078,000 


2.975_ 


2.733 


1,936,000 




after 5 min. 








after 5 min. 






Repeated 


2.18 


2.01 




Repeated 


3.019 
after i hour 

3.02q 
after 11 hours 






36,594 


after 5 min. 
2.254 
2.54 


2.08 


1,743,000 


50,684 


4.195 
in 10 minutes 


3.941 


1,448,000 




after 6 min. 














37.856 


2.894 




1,571,000 


Repeated 

(I 


4.226 

4.227 
in 7 hours 














(( 


4.227 
in 12 hours 














53,351 


Broke 















Hodgkinson on the tensile elasticity and permanent set of two 
wrought iron bars. The coefficients of elasticity E have been 
computed from the data contained in the first three columns as 
given by Mr. B. B. Stoney in his '' Theory of Strains in Girders 
and similar Structures." The following is the notation used : 



LI 

" Sets 
E 



pounds per square inch ; 

total elongation, or strain, for the bar ; 

permanent set ; 

coefficient of tensile elasticity = p x L 



- LI, 



Art. 32.] COEFFICIENT OF ELASTICITY. 219 

These experiments show some very interesting results. 

In the first place permanent sets were observed with the 
low intensities of stress of 8,003 ^^^^ 3,786 pounds, and it be- 
comes a question whether permanent sets would not have been 
observed with lower intensities and more delicate apparatus, 
at least for a short time after the material is subjected to 
stress. 

In both bars the largest value of ^ is found for the smallest 
intensity of stress. In bar No. I, the values of E decrease, 
with one exception, regularly from the greatest. In bar No. 
2, however, greater irregularity is observed ; there are two 
maxima, one for the intensity 1,262 pounds, and the other for 
about 12,000 with nearly regular gradations from these values. 

Considering the whole range in both bars, E may be con- 
sidered nearly constant until an intensity of about 24,000 
pounds per square inch is reached in each case ; it then begins 
to fall off very rapidly. 24,000 pounds per square inch, then, 
may be considered about the limit of elasticity for both bars. 

It is very important to observe the increase of strain with 
the lapse of time after the iinsit of elasticity has been consider- 
ably passed. 

Values of the coefficient of elasticity, therefore, mean little 
after that limit is exceeded. 

The results of the experiments on bar No. i are shown 
graphically in Fig. i. The values of "/ " are laid off vertically 
through 6^ to a scale of 20,000 pounds to the inch ; the tensile 
strains are the horizontal co-ordinates of the curve laid down 
at full size. The essentially straight portion of the curve 
between O and a is within what is ordinarily known as the 
*^ elastic limit." 

The equation for this portion of the line is : 

p = El', 
E being assumed constant if Oa is considered straight. 





220 
















I 


WROUGHT IRON IN 


TENSION. 












[Art. 


32 




































































































































































































































































































































































50000 








































































c 




- 








































































__ 


— 








. ■ 






_ 


















































^^ 




— 




' — 




































































-— 












































































■ 




















































40000 


















































































1 












\ . 




^ 


1 






























































( 






b 


\^ 


■^ 












































































^ 




' 














































































/ 


















































































300(X/ 


























































































































































































































































t^^ 








































































































































































20000 




























































































































































































































































































. 
















































































































































1000 















































































































































































1 
























































































/ 


































































































































































































1 


in 


:h 
















2 


in 


:he 


s 














3 


inche 


s 














4 


in 


:hes 





Fig.l 



The point ^ is at a vertical distance above O indicating 
about 24,000 pounds per square inch, t.e.^ about the elastic limit. 
Above this point the curvature of the line is very sharp, indicat- 
ing a rapid fall in the value of E and a rapid rise in the values 
of the strains /or L/. For "/ " = 27,000 (nearly) the table shows 
£ — 23,000,000 (nearly) and LI = 0.12 inch ; while for "/" = 
37,000, £ — 4,100,000 and LI = 1.095 inches (nearly). These 
phenomena are always characteristic of the limit of elasticity. 

Above the point d the curvature is slight, indicating (what 
the table shows) a comparatively slow change in the values of E. 

The table shows that bar No. 2 would exhibit a curve of 
precisely the same character but with a more rapid decrease to 
E above the elastic limit. The tests of this bar were not car- 
ried to failure on account of the breaking of one of the holding 
details. 



Art. 32.] 



COEFFICIENT OF ELASTICITY. 



221 



Within the elastic limit, the mean values of E may be taken 
about as follows : 



For bar No. i : 



For bar No. 2 : 



E = 28,000,000 pounds. 



E = 27,500,000 pounds. 



The next. Table IV., contains values of the coefficient of 
tensile elasticity (E) determined by Knut Styffe ('* The Elas- 
ticity, Extensibility and Tensile Strength of Iron and Steel," 
translated from the Swedish by Christer P. Sandberg). 



TABLE IV. 



KIND OF IROK. 



Hammered Bessemer Iron (square) 

Puddled, from Low Moor (round) 

" " Dudley " 

a ki ik Ik 

" " Motala, Sweden " 

" " " '■ (square) 

From Surahammar " 

o 

Swedish Rolled Iron from Aryd " 

ti ii t( (I II kk 

" " " " Hallstahammer (square) 



PER CENT 

OF 

CARBON. 



AREA OF 
SECTION, 



0.15 
0.20 
0.09 
0.09 
0.05 
0.20 
o. 14 
0.20 

I >:: [ 

0.18 
0.07 
0.07 



Sq. inch. 

o. 1003 
o. 1107 
0.1961 
0.1844 
0.2006 
0.1942 
0.1229 
0.2176 
0,1269 

0.2087 

0.2279 
o. 1891 

0.1965 



Inch. 

0.002 
0.001 
0.006 
0.008 
0.077 
0.008 



0.037 

0.003 
0.013 

O.OOI 



Pounds. 

32,320,020 
34,241,380 
31,976,020 
28,408,680 
27,448,000 
30,261,420 
29.575,220 
31,084,860 
30,467,280 

26,761,800 

27,791,000 
28,957,640 
30,810,380 



The '' Set " is the permanent elongation which '' the bar 
had just before the modulus {E) was taken." In the per cent, 
of carbon no distinction is here made between " in the bar 
tested " and " in the bars of the same kind," the two quanti- 
ties given by Styffe. 



222 WROUGHT IRON IN TENSION, [Art. 32. 

As a result of his experiments in regard to the effect of a 
change of temperature on the coefificient of tensile elasticity, 
he states (page 112 of the work above cited) : 

'' That the modulus (coefficient) of elasticity in both iron 
and steel is increased on reduction of temperature and dimin- 
ished on elevation of temperature ; but that these variations 
never exceed .05 per cent, for a change of temperature of 1.8"^ 
Fahr., and therefore such variations, at least for ordinary pur- 
poses, are of no special importance." 

In his " Physique Mecanique," page 58 of the " Premier 
M^moire," M. G. Wertheim gives three coefficients of tensile 
elasticity for wrought iron, each having about the value of 
29,680,000 pounds per square inch, and one for iron wire of 
about 26,474,000 pounds per square inch. 

Redtenbacher (Resultate fiir den Maschinenbau, Zweite 
Auflage, page 36) gives as the limits of the values of the co- 
efficient of elasticity, expressed in pounds per sq. in., about 
21,330,000 and 35,550,000. 

Reviewing the preceding values, therefore, it would appear 
that the coefficient of tensile elasticity for good wrought iron 
may be ordinarily taken to lie between 25,000,000 to 30,000,000 
pounds per square inch, with extreme values arising from vari- 
ation of mode of manufacture, chemical constitution, size of 
bar, etc., lying some distance either side of those limits. 

Since E = ■^, if / = i, /= -^ will be the elongation or 

tensile strain for each unit of stress ; hence, t/ie coefficient of 
elasticity is the reciprocal of the strain for a unit of stress. For 
an intensity of stress of 20,000 pounds, for example, then : 

, 20,000 , • 20,000 
/ = ■ — to 



25,000,000 30,000,000 



to 

1250 1500 



Art. 32.] ULTIMATE RESISTANCE. 223 

or a bar of wrought iron will be stretched : 

^ th to ^^ th 



1250 1500 

of its length. 

The coefficient of elasticity is thus seen to be a measure of 
the stiffness of the material. 



Ultimate Resistance and Elastic Limit. 

It has been found by experiment that bars of wrought iron 
which are apparently precisely alike, in every respect, except 
in area of normal section, do not give the same ultimate tensile 
resistance per sqiiare inch. Other things being the same, bars 
of the smallest cross section give the greatest intensity of ultiinate 
tensile resistance. 

Aside from the absence of uniform distribution of stress in 
the interior of the bar, as was shown in the section '* coefficient 
of elasticity,'' and the intensified effects of the processes of 
production on pieces with comparatively small cross sections, 
this result is to be expected from the circumstances which 
attend fracture. When a piece of material is subjected to 
tension to the point of rupture, not only a tensile strain of 
essentially uniform character, from end to end, takes place, but 
also a very considerable local transverse strain, or contraction, 
at the place of fracture. This latter manifests itself only 
shortly before rupture as a short '' neck " In the piece. Now a 
givQn percentage of '' local " contraction in the case of a large 
section involves a much larger absolute lateral movement of 
the molecules than In the case of a small section. But it is 
evident that this absolute lateral movement will exert a much 
more potent Influence toward severing the molecules suffi- 
ciently for rupture, than the percentage of contraction. Hence 



224 WROUGHT IRON IN TENSION. [Art. 32. 

the degree of local and lateral movement, required by rupture, 
will be reached with a less mean intensity of stress in the cases 
of large section than in those of small ones. But this is equiv- 
alent to a greater intensity of ultimate resistance for the small 
sections, and, as has been indicated, this conclusion is verified 
by experiment. 

The same considerations result in the additional conclusion 
that, other things being equal, the smaller sections will give the 
greater final contraction. But a greater intensity of ultimate 
resistance and greater final contraction involves 2. greater final 
stretcJi^ with the same length of piece. 

These last two conclusions will also be found to be here- 
after verified by experiment. 

Again, it is found independently of the effects of the pro- 
cesses of production, as might be anticipated, that the length 
in terms of the lateral dimensions of the test piece, within cer- 
tain limits, affects very perceptibly the ultimate resistance. 

If a specimen of the shape shown in Fig. 2 be broken by a 
tensile stress, it will, of course, fail in the reduced section MN, 
But before failure takes place, the reduced portion will be con- 

^ siderably elongated and the 
normal section correspond- 
ingly reduced, in conse- 
quence of the shearing 
"e strains in the oblique planes 
^'S'^ shown by the dotted lines. 

(See Arts. 3 and 4.) When the reduced portion in the vicinity 
of MN is very short in comparison with its lateral dimensions, 
it includes the whole of very few of these oblique planes, if any 
at all, consequently very little movement of these oblique 
layers over each other can take place ; in other words little 
or no reduction of section can take place before rupture. In 
this latter case, then, a greater area of metal section will offer 
its resistance to the external tensile force, at the instant of 
failure, than in the former, and a correspondingly greater in- 




Art. 32.] ULTIMATE RESISTANCE. 225 

tensity of ultimate resistance will be found. Thus the shape 
and dimensions of the test piece will considerably influence the 
ultimate resistance and strains, as will soon be shown by ex- 
perimental results. 

All the preceding conclusions, though given in connection 
with wrought iron, are independent of the nature of the ma- 
terial, and apply equally to steel and cast iron. 

Since the reduction of area of the fractured section and the 
elongation of the bar are true measures of the ductility of the 
iron, these are or should be always measured with care. 

Table V. exhibits in a very plain manner the decrease of 
ultimate tensile resistance with the increase of sectional area 
of round bars ; it is taken from the "Report of the Committees 
of the U. S. Board appointed to test Iron, Steel and other 
Metals, etc.," by Commander L. A. Beardslee, U.S.N. 

This decrease is probably partly due to the effect produced 
upon the iron by the rolls as it passes through them ; the 
bars of smaller sections being more " drawn," and at a lower 
temperature in consequence of the lesser mass cooling more 
quickly. 

The notation of the table is the following : 

" Diay = diameter of the round bar in Inches ; 
'< T." = ultimate tensile resistance ; 
^^ £. Z." == elastic limit. 

It will be observed that the ultimate resistance per square 
inch varies between widely separated limits, in some cases, for 
the same diameter of bar. This is due to the fact that the 
different bars, even of the same diameter, were from a number 
of different mills, and consequently involved different treat- 
ment in manufacture, chemical constitution, etc. A general 
view of the table, however, shows in a marked and satisfactory 
manner the decrease of 7" with the increase of the diameter or 
area of normal section. The last fourteen bars of the table 



226 



WROUGHT IRON IN TENSION, 



[Art. 32. 



are of the same manufacture, and show a decrease in T as 
nearly uniform as could be expected. 

TABLE V. 

Ultimate Resistance and Elastic Limit in Pounds per square inch of 

Original Nor??tal Section. 



DiA. 


T. 


E.L. 


DiA. 


T. 


E.L. 


DiA. 


T. 


E.L. 


I< 


59,885 


40,980 


I?^3 


53.016 
51,296 


35,379 




50,969 


30,814 


H 


54,090 




31,992 


(i 


50,307 


29,767 


<f 


62,700 
59,000 




iH 


50,594 
57,052 


34,940 
38,417 


i/s 


48,953 
55.803 


31,031 




57,700 




" 


56,505 


32,496 


u 


53,100 


32,074 




55.400 






55,131 


33,771 


i( 


52,875 


35,641 




52,275 


39,126 


" 


54,540 




(( 


52,505 


32,312 




55,450 




" 


55,415 


32,869 


" 


51.459 


27,816 




52,050 
57,660 




i( 


54,354 
5'^--544 


34,617 
33,027 


(( 


50,363 
51,039 


33,067 




51,546 

50,630 


35,933 
33,931 


t( 


53.512 
52,819 


34,840 


;t 


49,744 
48,670 


35,615 
23,250 




61,727 




" 


52,736 


3-*'9oi 


2.0 


60,213 


31,441 




57,363 
57,807 


37,415 
39*230 


i( 


52,700 

52,155 


35.880 
27,708 


«( 


52,914 
49,164 
51,684 


31,198 




56,790 


36,885 


t( 


51,994 


32,054 


" 


33,104 




51,921 


31.300 




51,456 


34.591 


" 


52,127 


32,461 


IH 


52,Si9 
51,400 
60,458 


32,267 
34,600 
37,344 


J% 


51,047 
56,344 

57,402 


35,889 
35,701 


(( 


52,011 
51,146 
50,000 


34,702 
28,567 
36,184 




57,470 


31,900 


" 


56,227 


33,207 


" 


50,1.71 


28,983 




57,498 


41,311 


" 


54,334 


32,163 


u 


47,812 


35,864 




55,927 


37,250 




53,339 


33,540 


(( 


48,249 
46,151 


31.413 
36,050 




54,644 


34,695 


" 


53,614 


30,664 


(( 




53,900 


26,787 


a 


52,675 


33,745 


2-J- 








53,035 


34,410 


" 


52,314 


29,364 


'^l (J 


51,559 






52,267 


32,019 


" 


52,401 


34.012 


^\y 


49,422 
50,481 




^% 


59^461 


36,501 


" 


51,205 


33,318 


2% 






57,897 


32,469 


" 


50,970 


33,625 


(( 


51,225 
48,382 

51,666 
51,530 






5^,782 
56,334 
55,253 


35,596 
33,921 
34,784 




56,595 
54.114 
57,789 


38,310 
34,160 




30,459 




53,893 
53,247 
53,752 
52,970 


32,712 
32,520 

32,075 




57,874 
54,410 
53,846 
55,018 


31,354 
36,573 
34,283 


2K 


40,290 
48,898 
46,866 

48,475 
47,428 


32,163 

28,241 
28,932 




53,022 




" 


53,264 




<( 


29,941 
20,758 




50,040 


30,730 


u 


53,154 
51,509 
50,395 


35,323 
29,404 

36,254 




47.344 


T% 


58,026 
58,021 


37,348 
32,152 


It 


2% 

3-0 

-.1/ 


46,446 
47,761 


26,333 
26,400 




54.949 


31,030 


" 


50,547 


35,954 


Iv 


47,014 


24,591 
24,061 




54,277 


33,622 


(( 


49,8i6 
50,129 

56,577 


31,214 
32,271 


^6 


47,000 




52,733 


34,6c6 j 


i( 


3K 


46,667 


23,636 




53,557 


33,650 


1-13. 




4.0 


46,322 


23,430 




52,537 


34,469 


^1 b 











In the words of the Report, as given by Wm. Kent, C.E., in 



Art. 32.] 



UL TIM A TE RESISTANCE. 



227 



the abridgment, *' The elastic limit as given is not from per- 
fectly accurate data ; it is simply the amount of stress which 
produced the first perceptible change of form, divided by the 
bar's area." 

TABLE Va. 



Rectangular Bars. 









STRESS IN LBS. PER SQ. IN. 


PER CENT. OF 


NO. 


KIND OF IRON. 


BAR. 


Elastic 




Final elongat'n 


Final contrac- 








Limit. 


Ultimate. 


in 80 inches. 


tion. 






Inches. 










I 


Single Refined 


3x1 


29,000 


52,470 


18.0 


31.0 


2 


Double " 


3 X I 


31,000 


53,550 


16.0 


27.7 


3 


Single " 


5 X li 


27,330 


50,410 


16.6 


24.1 


4 


Double " 


5 X li 


27,170 


50,920 


19.0 


25-7 


5 


Single '' 


3 X I 


28,330 


48,700 


I3-I 


27.1 


6 


Double '' 


3x1 


29,170 


51,370 


22.2 


35-6 


7 


Single " 


5 X i| 


24,830 


49,240 


16.0 


18. 1 


8 


Double " 


5 X li 


27,170 


51,010 


19.7 


29-5 



Table V^;. shows the results of some tests in the U. S. 
Govt, machine during 1 88 1, at Watertown, Mass. Nos. i and 
2 are means of four tests ; the others are means of three. Nos. 
I, 2, 3 and 4 are for bars from the Elmira Iron and Steel Roll- 
ing Mill Co.; Nos. 5, 6, 7 and 8 are from the Passaic Rolling 
Mill Co. As a rule the large bars give the least elastic limit 
and ultimate resistance. 

It is also important to observe that the double refined iron, 
with two exceptions, gives the highest results of all kinds. 

It appears from an examination of the tables that the elas- 
tic limit varies, approximately, from a half to two-thirds the 
ultimate resistance. 

The ultimate resistance, it is to be particularly observed, is 



228 WROUGHT IRON IN TENSION. [Art. 32. 

given in pounds per square inch of original sectional area. On 
account of the reduction of the fractured section, the ultimate 
resistance should be specifically referred either to its own sec- 
tion (to be noticed hereafter) or to the original section. The 
customary reference is to the latter, though it is frequently 
interesting and important to make an accompanying reference 
to the former. 

TJie influence of the reduction of the piles between the rolls 
was next examined by the same committee. It was found that 
the additional working involved in the increased reduction of 
the pile, as it passes through the successive rolls, in the process 
of manufacture, considerably increases both the ultimate re- 
sistance and elastic limit. Tables VI. and VII., condensed 
from those containing the results of the committee's experi- 
ments, show this effect in a very satisfactory manner. The 
notation is as follows : 

D — diameter of bar in inches ; 

A = area of normal section of original pile in square 

inches ; 
Per cents. = area of bar in per cent, of area of pile ; 
T = ultimate tensile resistance in pounds per square 

inch of entire bar ; 
T' = ultimate tensile resistance in pounds per square 

inch of core of bar ; 
E. L. = elastic limit in pounds per square inch of entire 

bar ; 
E'. L\ = elastic limit in pounds per square inch of core 

of bar. 



As is to be anticipated in such cases, some irregularities 
are exhibited in the tables, but they are very few, while the 
general result is unmistakable. On the whole, a considerable 
increase in the values of T is observed in connection with 



Art. 32.] 



INFLUENCE OF ROLLS. 



229 



a decrease in the values of '' Per cents.'' Values of the elastic 
limit show greater irregularities. 

TABLE VI. 

Comparison of the Reductiojts by the Rolls, with the Effects upon 
Tenacity, and Elastic Lijnit of Round Bars. 



D. 


A. 


PER CENTS. 


T. 


T'. 


E. L. 


E'.L'. 


4 


80 


15 70 




46,322 




23,430 


3^ 


80 


12.03 




47,000 




24,961 


3 


80 


8.83 




47,761 




26,400 


2\ 


80 


6.13 


47,344 


47,428 


29,758 


29,941 


2 


72 


436 


47,872 


48,280 


35,864 


31,892 


I| 


36 


6.68 


50,547 


48,792 


35,954 


38.992 


i4 


36 


4.90 


50,820 


51,838 


35,087 


36,467 


li 


36 


3-41 


52,729 


49,Soi 


39,608 


40,534 


I 


25 


3-H 


51,921 


51,128 


39,066 


38,596 


f 


I2i 


3 60 


50,673 


50,276 


33,933 


35,933 


\ 


9 


2.17 


52,275 


52,775 


38,445 


39,126 


\ 


3 


1.60 


57,000 


59,585 


Lost 


Lost 



TABLE VII. 

Another Table sJiowing Similar Results, with T' and E' L' , for 

Core, omitted. 



D. 


A. 


PER CENTS. 


T. 


E. L. 


2 


27 


11.63 


51,848 


32,461 


\\ 


15 


11.78 


53,550 


34,090 


A 


27 


10.22 


54,034 


33,610 


If 


15 


9.90 


54.277 


33,622 


M 


27 


8.90 


55,018 


34,283 


14 


15 


8.18 


56,478 


33,251 


If 


27 


7.68 


56.344 


35,889 


^^. 


15 


6.62 


56,143 


32,267 



The opinion of the committee on the effect of iinderheating 



230 



WROUGHT IRON IN TENSION. 



[Art. 32. 



or overheating is thus given in the abridgment of their report 
by Wm. Kent, M.E. : " The indications are that if a bar is 
underheated it will have an unduly high tenacity and elastic 
limit, and that if overheated the reverse will be the case." 

In the words of the report : *' The evidence submitted is of 
sufficient value to justify us in asserting that variations in the 
amount' of reduction by the rolls of different bars from the 
same material produce fully as much difference in their physi- 
cal characteristics as is produced by differences in their chemi- 
cal constitution." 

The committee also made some valuable experimental in- 
vestigations with the object of ascertaining the influence of the 



7 



\ 



Fig. 3 



Fig. 4 



relative dimensions of the test piece, already remarked upon in 
connection with Fig. 2. Eighteen specimens were prepared, 
of which Figs. 3 and 4 represent types. 

Fig. 3 represents a specimen whose middle portion is turned 
down to a uniform diameter. Seventeen of the specimens 
were of this kind, with lengths of cylindrical portions varying 
from J^ inch to 10 inches. Fig. 4 represents the eighteenth 
specimen with simply a groove in the centre, in which, at ab^ 
the fracture took place. In this latter specimen, the reduction 



Art. 32.] 



INFLUENCE OF LENGTH. 



231 



of area at the section of failure must necessarily be touch less 
than in those like Fig. 3 ; hence, the ultimate resistance v/ill be 
correspondingly greater. 

Table VIII. is taken from the report already cited, and 
contains the results of the experiments on the eighteen speci- 
mens prepared in the manner indicated above. 

L = original length in inches ; 
/ = per cc7it. of elongation ; 
a = per cent, of contraction of fractured area ; 
t = stress in pounds per square inch at first stretch ; 
'T = ultimate tensile resistance in pounds per square 
inch of original section. 

TABLE VIII. 



NO. 


L. 
10 


/. 


a. 


/, 


T. 


REMARKS. 


I 


23. 1 


38.2 


29,678 


54,888 


Slight seam. 


2 


9:^ 


24-3 


36.5 


28,011 


55,288 




3 


9 


21 5 


311 


29.345 


55.355 




4 


%\ 


22.0 


31 .2 


29.345 


55.622 




5 


n 


25.0 


39-9 


30,840 


54.890 


Slight seam. 


6 


7 


25.8 


38.6 


30,412 


55,488 




7 


6i 


22. 1 


40.0 


28,562 


51,800 


Bad seam. 


8 


6 


22.3 


34-7 


30,600 


55,418 




9 


5^ 


25-4 


39-3 


29,475 


55,333 




10 


5 


21.2 


32.2 


29,278 


55,887 


Slight seam. 


II 


4 


25-7 


37-4 


29.705 


55,532 




12 


/1 1 
j2 


26.7 


36.6 


31,817 


55,482 




13 


3 


27.0 


38.3 


31.123 


56,190 




14 


2 


27.0 


36.2 


33,428 


56,428 


Seamy. 


15 


1} 


26.0 


34-0 


42.249 


57,096 


<< 


16 


I 


37-0 


34 3 


34,288 


58.933 




17 


i 


30.0 


37-9 


57,565 


59,388 


Seamy. 


18 


Groove 




20.6 


45,442 


71,300 





The diameters at the section of failure were nearly uniform 
and originally about 0.97 inch. 



2^2 WROUGHT IRON IN TENSION. [Art. 32. 

The values of /, a and 7" are as nearly uniform as could be 
expected until the length decreases to about 4 diameters 
(2 inches). 

For the grooved specimen / and T are very large, and a 
very small. 

Other experiments on a still softer iron were made with 
the same general results. 

*' In conclusion," states the committee, " our results lead 
us to the decision that, in testing iron, no test piece should be 
less than one half inch in diameter, as inaccuracy is more prob- 
able with a small than with a large piece, and the errors are 
more increased by reduction to the square inch ; that the 
length should not be less than four times the diameter in any 
case ; and that, with soft, ductile metal, five or six diameters 
would be preferable." 

In Vol. II. of the '' Transactions of the American Society of 
Civil Engineers," Mr. C. B. Richards has given a paper in 
which are recorded the results of some experiments exhibiting 
the influence of the relative dimensions of the specimens. The 
average of eight tests of Burden's ''best" iron, with "long" 
specimens (similar to Fig. 3) varying from 5 to 5^^ inches in 
length and 0.62 to i.oo inch in diameter, gave : 

T = 49,588 pounds ; a = 46.7 per cent. ; 

/ = 30.4 per cent. 

With '' short " specimens (like Fig. 4) of the same iron, the 
average of six tests gave : 

T = 62,089 pounds ; a ~ 29.5 per cent. 

The large value of T and small value of a, for the '' short " 
specimens, are thus seen to be very marked in contrast with 
the same quantities for the "long" specimens. 



Art. 32.] INFLUENCE OF SKIN, 233 

Other experiments of Mr. Richards, showing the same re- 
sults, will be given in connection with the resistance of boiler 
plates. 

It has long been the impression that there exists a consid- 
erable difference between the ultimate tensile resistance of the 
" skin " of a bar of iron and that of the portion of the bar un- 
derneath the skin. The U. S. committees, therefore, broke a 
number of bars first with the skin on, or '' in the rough," and 
then with the skin turned off. In a large majority of the cases, 
the rough bars gave the highest ultimate resistance per square 
inch, by a small amount, while in a few cases the results were 
of the opposite character. On the whole, however, " the ac- 
cumulated evidence indicates that the strength of the skin of 
the bar is greater in proportion to its area than that of the rest 
of the bar." 

All the tests, of which the results have hitherto been given, 
were made on round bars, or on specimens turned from them. 
Results of tests on other iron will now be detailed, and it will 
be convenient to use the following and customary symbols for 
the various kinds of '^ shape " irons : 



L 


for angle irons : 




J. 


for tee irons ; 




C, 


for channel bars ; 


. 


I 


, for eye beams ; 




D, 


for rectangular bars or 


" flats " ; 





for rounds ; 




+ , 


for star sections ; 





in short, any shape iron, or steel, is represented by a skeleton 
of its section. 

Table VIIL?. contains the results of tests by Henry Wood, 
Inspector for the O. M. O. & O. Railway of Canada, on some 
bridge members at Phoenixville, Penn., in Oct., 1878. 



234 



WROUGHT IRON IN TENSION. 



[Art. 32. 



TABLE Villa. 



SECTION. 


LENGTH. 


E.L. 


T. 


CONT. 


STRAIN. 


REMARKS. 


Inches. 


Inches. 












3^ X lA- 


338 


31,000 


56,414 


0.28 


0.14 


■ Flats. 

' Not broken. 


3^ X I 


264 


32,000 


54,444 


0.40 


0.15 


3 X li 


336 


29,000 


61,331 


0.09 


0. II 


3^ X Si 


80 


30,000 


56,474 


0.27 


0.15 


ifo 


320 


31,900 


57,530 


0.30 


0. 13 


1 


i|o 


80 


31,800 


56,454 


0.43 


0.19 


> Rounds. 


Ifo 


254 


27,000 


56,351 


0.50 


0.17 


^ 



E. L. — elastic limit in pounds per square inch. 

T. — ultimate resistance in pounds per square inch of 

original section. 
Cont, — reduction of original area. 



Strain — stretch of " LengtJi' 



The iron in these bars was that of the Phoenix Iron Co. 

It should be stated that the tests were made in a lever 
machine in which the friction may have amounted to 4 or 5 
per cent, of T. 

The following, Table IX., contains the results of some ex- 
periments made in 1875 by Mr. G. Bouscaren for the Trustees 
of the Cincinnati Southern Railway, and is taken from the 
'' Report on the Progress of Work, etc.," by Thomas D. Lovett, 
Con. and Prin. Engr., published in 1875. 

The tests were Txiade on eye bars, and only such have been 
selected for the table as broke in the body of the bar and at a 
section out of reach of the influence of the fire used in the 
process of forming the head. 

Nos. 2 to 18 are corrected for the error of a spring gauge, 
as is indicated in the report cited. 



Art. 32.] 



EYE BARS. 



235 



TABLE IX. 


NO. 


BRAND. 


LENGTH. 


SECTION. 


T. 


STRETCH. 






Ft. In. 


Inches. 




Per cent. 


2 


D. S. 


10 9i% 


3 >< i b 


50,667 


20 


6 


P. 


10 9A- 


3 X f 


47,734 


7 


8 


D. S. 


5 


3 X f 


50,220 


10 


9 




5 


3x1- 


50,220 


13 


10 




5 


3x1 


50,220 


14 


II 




5 


3 X f 


48,980 


II 


12 




5 


3x4 


51,556 


14 


13 




5 


3x4 


48,980 


10 


14 


P. 


5 


3 X 1 


53,870 


19 


15 




5 


3x1 


53,870 


18 


16 




5 


3 X i 


49,500 


16 


17 




5 


3 X 1 


50,666 


15 


18 




5 


3 X i 


47,380 


11 


26 




5 


3x4 


53-700 


16 


29 


D. S. 


5 


3 X f 


52,600 


13 


30 




5 


3 X f 


50,600 


13 


31 




5 


3 X f 


52,600 


14 


32 




5 


3 X 1 


48,100 


10 


33 




5 


3 X 1 


50,600 


15 


35 


P. 


5 


3x1 


52,000 


15 


36 




5 


3 X 1 


52,900 


14 


37 




5 


3 X f 


50,900 


7 


38 




5 


3 X f 


55,100 


13 


40 


D. S. 


5 


3 X f 


48,100 


14 


41 




5 


3x1 


42,400 


14 


42 




5 


3 X f 


43.800 


17 


43 




5 


3 X f 


49,800 


T5 


45 




5 


3 X f 


49,200 


16 


48 




5 


3x4 


4^,900 


15 


53 




5 


3 X I 


47,500 


16 



The following is the notation : 



^'- Brand''' :• ■< 



D. S. represents Diamond State Works, 
Wilmington, Del. 

P, represents Pencoyd Works, Philadel- 
phia, Penn. 
'' Length " — distance from centre to centre of eyes. 



236 WROUGHT IRON IN TENSION, [Art. 32. 

'' Section " represents dimensions in inches of rectangu- 
lar section of bars. 

T. = ultimate tensile resistance in pounds per square 
inch of original section. 

*' Stretch " represents per cent, of original length. 

Table X. has been written by the aid of the same report as 
that from which the preceding was taken. 

It contains the records of tests made on specimens sup- 
plied, under specifications, by various bridge building com- 
panies from the mills indicated in the column '■^Brandy With 
the exceptions of Nos. 15, 16 and 17, the test specimens were 
turned ij^ inches round ; those three were of rectangular sec- 
tion and of the dimensions given in the column '* Diauh' 

It is stated in the report that the ultimate resistances are a 
little too large, as they involve the friction of the plunger in 
the hydraulic cylinder of the testing machine. 

The notation is the following : 



^ " P.," Phoenix Iron Co. ; 
'' U, /.," Union Iron Mills ; 



"■Brand'' : ■< 



'' a R.;' Cleveland Rolling Mills ; 

'' O. F.;' Ohio Falls Iron Works ; 

" D. 5.," Diamond State Works ; 

"■ K. /.," Kellogg Iron Works. 
'^ Dia.'' = diameter, or dimensions, in inches, of original 

section. 
" T.'' = ultimate tensile resistance in pounds per square 

inch of original section. 
" Strain " = elongation for eight inches in length. 
" Conty = contraction of original section. 



'fc>' 



The two latter were measured after failure took place ; they 
have been computed from the data given in the report. 



Art. 32.] 



SPECIMEN TESTS. 



^?>7 



TABLE X, 



Tests of Speciinen Bars of Iron, Siibniiitcd iviih Bids for Ohio River 



Bridge. 



NO. 


BRAND. 


DIA. 


T. 


STRAIN. 


CONT. 


REMARKS. 


I 


P. 


1.25 


54,400 


0.31 


0.45 


) I bar bent 180° without 


2 


P. 


1.25 


=;4.400 


0.29 


0.41 


f fracture. 


3 


U. I. 


1.25 


53,800 


0.31 


040 


\ 2 bars bent 180° with- 


4 


U. I. 


1.25 


53,800 


0.28 


0.37 


) out fracture. 


5 


K. I. 


1.24 


54,600 


0.30 


0.39 


) I bar bent 180° without 


6 

7 


K. I. 


1.25 

1.25 


53.400 
55,100 


0.29 
0.21 


0.41 
0.20 


j fracture. 






8 




1.25 


55,500 


0.31 


0.38 








9 


C. R. 


1.25 


53.800 


0.17 


0.15 




10 


C. R. 


1.25 


54,200 


0.19 


0.21 


'i bar bent 180° without 


II 
12 


C. R. 
C. R. 


1.24 
1.24 


60,800 
60,800 


0.28 
0.266 


0.31 
0.27 


fracture. 
^ I bar bent 135". Frac- 
ture showed partly 
crystalline. 


13 
14 


U. I. 
U. I. 


1.25 

1.25 

(2.59) 


52,000 
50,400 


0.27 
0.28 


0.36 
0.33 


>• I bar bent 135°. 


15 


U. I. 


^ ■ 


51,700 




0.08 












(0.49) 










16 


U. I. 


Yt\ 


49,400 




0.09 




) 










( 0.48 














2.64) 










17 


U. I. 


] ^ r 


52,400 




O.II 












(0.37) 










18 
19 


0. F. 
0. F. 


1.25 
1.25 


51,600 
52,200 


0. 13 
0. 13 


0.09 
0.09 


j- 2 bars bent 135°. 


20 


D. S. 


1-25 


53,000 


0.30 


0.43 


) 2 bars bent 180" with- 


21 


D. S. 


1.25 


52,100 


0.28 


0.42 


) out fracture. 



In regard to the column of remarks the report says : " One 
or two of the specimen bars furnished by each bidder were 
tested by being bent cold under the hammer until fracture, and 
the result is noted in the column * Remarks' " 

These tests were made in 1875. 



238 



WROUGHT IRON IN TENSION. 



[Art. 32. 



The next table is again taken from the same report, and 
contains the results of tests made on specimens taken from 
plates, angles, rods. Keystone column iron and rivet iron. "7"," 
as usual, represents the ultimate tensile resistance in pounds 
per square inch of original section. 

TABLE XL 



NO. 


SPECIMEN. 


T. 


REMARKS. 




Inches. 






I 


0.86 X 0.40 


50, 100 


Specimen from 14I x f plate. 


2 


0.88 X 0.40 


48,900 






' 14I X f ' 




3 


0.86 X 0.36 


45,400 






10 X g * 




4 


0.84 X 0.38 


41,400 






' 10 X f ' 




5 


0.86 X 0.26 


49,200 






' 10 X i ' 




6 


0.90 X 0.26 


44,400 






' 10 X 4 ' 




7 


0.62 X 0.38 


56,000 






* 10 X ' 




8 


0.64 X 36 


49,900 






10 X i ' 




9 


0.62 X 0.40 


54,600 






14I X 5 ' 




10 


0.62 X 0.42 


52,600 






' 14I X f ' 




II 


. 84 X . 40 


52,600 






' • i\ -x 2.\ angle iron. 


T2 


0.54 X 0.42 


53,800 






24 X 2i " 


13 


l[§ X I 


46,700 






' i}-^ X I bar. 


14 


if Diam. 


55,200 






* if rod. 


15 


li " 


58,100 






' lii rod. 


16 


0.84 X 0.26 


47,700 






' 6" Keystone Column. 


17 


0.84 X 0.30 


49,000 






' 8'^ 


18 


1 Diam. 


48,200 






' 1" rivet for column. 


19 


% " 


46,900 






' f <' " 



Nos. I, 3, 5, 7 and 9 were taken from the edges of the 
plates, while Nos. 2, 4, 6, 8 and 10 were taken from the middle 
of the same. It is important to observe that the specimens 
from the edge, in every instance, gave considerably higher 
values of 7" than those from the middle. 

These tests were also made in 1875, and as they were com- 
pleted in a Riehle machine, they may be confidently considered 
exact. All the iron was rolled at the Union Iron Mills of Pittsr 
burgh, Penn. 



Art. 32.] 



PLATE SPECIMENS. 



239 



The same report will again be drawn on for Table XII. of 
" Tests of Tensile Strength of Iron furnished (in 1875) by the 
Baltimore Bridge Co., for Kentucky River Bridge." 



TABLE XII. 



u 



I 

2 

3 
4 
5 
6 

7 
8iW 

9 
10 
II 
12 
13 
14 
15 
16 

17 

18 

19 



W. W. 



u 



I. 
w. 



D. 



0.88 

0.98 

0.75 

0.75 

0.75 

0.375 

0-375 

0.80 

0.72 

075 
0.75 
0.75 
0.75 
0.75 

0-75 
1. 00 
1. 00 
1. 00 
1. 00 



X o. 
X o. 
X o. 
X o. 
X o. 
X o. 
X o. 
X o. 
X o. 
X o. 
X o 
X o. 
X o. 
X o. 
X o. 
X o. 
X o. 
X o 
X o 



26 
26 

so 

50 

50 

75 

75 

50 

•52 

•50 

48 

•50 

•50 

•50 

•50 

.31 

•31 

•25 

•25 



T. 



48,5oo|o, 

49,i00|O. 

47,7000. 

46,goo[o 

45,300 

50,100 

54,400 

46,000 

42,900 

46,400 

48,8oolo 

49,500- 

48,000 — 

45,3000 

47,90o|- 
49.5oo|o 
50,600 o 
56,000 o 
56,5000 



10 



16 
14 
13 
09 



0.25 
0.17 
0.22 
0.12 
0.22 
0.12 
0.16 
0.22 
0.13 
0.22 
0.21 



o.W 



0.29 
0.26 
0.17 
0.08 



Specimen from edge 

middle 

edge 

edge 

middle 

edge 

middle 

edge 

middle 

edge 

middle 

edge 

middle 

edge 

middle 

edge 

middle 

edge 

middle 



f 12 X J: 

12 X 4 

12 X I 

12 X I 

12 X i 

16 X I 

16 X f 

16 X I 

16 X i 

16 X I 

16 X i 

16 X 4 

16 X 

18 X 

18 
19 
19 
24 

24 X 



plate. 



4 
\ 

'I b 
5 

] 6 
I 

4 
JL 



The notation is as follows : 



'^ Brand' 



A 



f' W, W.;' Wilson Walker & Co., of Pitts^ 
burgh, Penn. 
'' U, /.," Union Iron Mills of Pittsburgh, 
Penn. 

*' D.'' = dimensions of sections of specimens in inches. 
" 71" .— ultimate tensile resistance in pounds per square 
inch of original section. 



240 



WROUGHT IRON IJV TENSION. 



[Art. 32. 



" Strain " — elongation, at failure, for three inches of 

length. 
** Coiity = contraction, at failure, of original section. 



Nos. 18 and 19 were taken from the middle of the plate, but 
all the others were taken from *' crop " ends. 

The report states that the percentages are only approxi- 
mate, as they are based on measurements made by an ordinary 
rule. It also states that specimens showed fracture, on bend- 
ing cold, between 90° and 180°. 

It is to be observed that the greater resistance shown by 
the specimens taken from the edges of the plates, in the pre- 
ceding table, does not, as a rule, appear in this one. 

It is inferred, though not definitely stated in the report, 
that all specimens of plates were tested in the direction of the 
fibre, that being the direction in which the actual plates would 
be directly stressed in the structures contemplated. 

TABLE XIII. 



SIZE OF BAR. 


DIAMETER. 


T. 


£. L. 


STRAIN. 


CONT. 


Inches. 


Inches. 










4 X If 


1. 00 


56,200 


31,600 


0.26 


0.33 


4 X i|- 


1. 00 


54,600 


26,700 


0.25 


0.31 


4i X Is 


1. 00 


61,100 


31,600 


0.24 


0.28 


4i X i| 


1. 00 


59.500 


30,000 


0.17 


0.28 


4^ X If 


1. 00 


56,600 


27,000 


0.29 


0.31 


4^ X If 


1. 00 


56,200 


30,000 


0.20 


0.36 


6x1+ 


1. 00 


56,200 


28,300 


0.32 


0.41 


6 X I.5r 


I. GO 




28,326 













Table XIII. contains the results of tests made by the writer 
at Phcenixville, Penn., in 1881. The test specimens were all 
one inch in diameter and turned to a uniform section for a 
length of ten inches ; they were taken from Phoenix '' Best 



Art. 32.] BOILER PLATE. 24I 

Best " bars of the sizes given in the extreme left vertical col- 
umn. The column *' E. Z.," elastic limit in pounds per square 
inch of original area, contains values based on measurements 
made as accurately as possible with a pair of fine dividers and 
a scale graduated to hundredths of an inch. The results, 
therefore, can only be considered as loosely approximate. 
The remaining notation is same as used in the preceding 
tables. 

The failure of the clamps to hold the last specimen pre- 
vented a complete record. 

The " Strains " are for six inches of length. 

It should be stated that the testing machine was a lever 
one, and the friction may have amounted to five per cent. 

In all the preceding tables the length for which the ^^ Strain " 
is given should be carefully borne in mind. A considerable 
" local " strain takes place at the section of fracture, which 
causes the per cent, of elongation, or strain, to be much greater 
for a very short length than for a longer one. 



WrougJit-Iron Boiler Plate. 

The " Report of the Committee of the Franklin Institute," 
made in 1837, has already been cited on page 215. That report 
contains the results of a great number of experiments made on 
boiler plates of both wrought iron and copper. A very few 
of the values of the ultimate tensile resistance, Z, in pounds 
per square inch, are given in Table II. of this Article, and 
reference may be made to it. 

Table XIV. contains the limiting values of T from the 

tables of the report whose numbers are given in the extreme 

left hand column. The tables selected are such as to give 

fairly representative results, and are chosen for no other 

reason. 

16 



242 



WROUGHT IRON IX TENSION; 



[Art. 32. 



TABLE XIV. 



TABLE. 


KIND. 


LIMITING VALUES OF T. 


TEMPERATURE. 


FIBRE. 


XXXI. 

XXXII. 

XXXVI. 

XLVII. 

LII. 

XL. 

L. 

LI. 


H. 
H. 
P. 


53,543.00 to 66,500.00 
34,990.00 to 48,041.00 
48,308.00 to 73,385.00 
54,361.00 to 78,000.00 
44,149.00 to 65,897.00 
41,734.00 to 65,141.00 
55,869.00 to 65,700.00 
54,442.00 to 62,709.00 


About 65° Fahr. 

.. 75° " 
54° to 580° 'i 
48' to 88° " 
80° to 580° " 
66° to 90° " 
About 70° " 
70° " 


Across. 
Along. 


Ens^lish. 

H. 
Rolled. 


< ( 

< ( 

Across. 



*'i7." signifies ''hammered into slabs and rolled.' 
'' F.'' signifies " puddled and rolled." 



In Table XXXVI. tJie highest values of T occurred at the 
highest temperatures. 

In Table LIT, the lowest values of T occurred at lowest 
temperatures. 

The results in Table XIV. v/ere recorded from the tests of 
*' long " specimens but of very small area of normal section, 
i.e., from about o.io square inch to 0.20 square inch. 

This committee made numerous experiments to determine 
the resistance of boiler plate in different directions in reference 
to the fibre of the iron. The results were by no means of a 
uniform character. In one set of forty strips cut in each direc- 
tion (along the fibre and across it), the length strips showed an 
excess of resistance varying from one per cent, to twenty. This 
comparison was made principally on the minimum resistance 
of each bar, but the committee state that the result would not 
have been much different if the mean had been taken. 

On reviewing all their experiments, the committee con- 
cluded that lengthivise of the fibre, the boiler iron zuhich they 
tested was about six per cent, stronger than across the fibre. 



Art. 32.] 



BOILER PLATE. 



243 



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M 


M 


a 


IT" 

N 


d 6 




















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I— I 

X 

H 



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244 WROUGHT IRON IN TENSION. [Art. 32. 

They also determined that the weakest direction of all was 
diagonally across the fibres, but their experiments did not en- 
able them to determine quantitative results. 

Table XIV<^. is taken from the ''Transactions of the Ameri- 
can Society of Civil Engineers," Vol. II. It contains the re- 
sults of some experiments on several different kinds of plate 
iron by C. B. Richards, M.E., and among other things it reveals 
the difference between " long " and " short " specimens. 

Column 'W^." shows the number of tests, of which Txs 
the average ultimate tensile resistance in pounds 
per square inch, T' the highest and T^ the lowest, 
all being referred to the original section. 

Column " Conty shows per cent, (of original section) of 
contraction at section of failure. 

Column " Strain *' shows per cent, (of original length) of 
elongation. 

Column ^^ Spec.'' shows kind of specimen, i.c.^ "long" or 
" short," also direction of stress in reference to fibre ; 
^^ LL'' signifies "long and along fibre;" "Z(7" 
"long and across fibre;" while "vSZ"and "5(7" 
signify " short and along " or " across fibre," respect- 
ively. 

In column ''Brandt' "Z?. 5." signifies "Bay State;" 
"^. 5. Hr ".Bay State Homogeneous Metal;" 
" rr "Thorneycroft," English ; ''Penny "Pennsyl- 
vania ; " " 6'. Fy " Sligo Fire Box.' 

Different brands of the same make, though given by Mr. 
Richards, have been neglected. 

The lengths for which the " Strains " existed are not given, 
although they should be. The long specimens were three or 
four inches between the shoulders. 

In his "Treatise on the Resistance of Materials," Prof. De 



Art. 32.] INFLUENCE OF ANNEALING. 245 

Valsen Wood gives the following results of some boiler-plate 
tests at the shops of the Camden and Amboy R. R. by Mr. F. 
B. Stevens. 

" Av. breaking weight in pounds per square inch. . . . 54,123.00 

Highest " " '' '' " " " 57,012.00 

Lowest^' " '' '' '' '' " 51,813.00" 

The experiments of Sir Wm. Fairbairn on English boiler 
plate (*' Useful Information for Engineers, First Series," p. 259) 
along and across the fibres, gave irregular results, but other 
English experiments of Easton and Anderson would seem to 
make the resistance across the fibres from 5 to 15 per cent, less 
than that along the fibres. 



Effect of Aitnealing, 

The Franklin Institute Committee determined the effect of 
annealing, at different temperatures, on about 56 specimens of 
boiler plate and wire iron. Table XV. is condensed from that 
giving their results on boiler plate. 

The mean value of 7^ for five specimens of iron wire 0.19 
inch in diameter, before annealing, was : 

T = 73,830. 

After annealing by heating to redness and cooling in dry 
ashes, the mean of five specimens was : 

T' = 58,101. 

After annealing at red heat and quenching in water, the 
mean of another five specimens was : 

T' = 53,578. 



246 



WROUGHT IRON IN TENSION. 



[Art. 32, 



TABLE XV. 











DECREASE 


NO. 


T. 


ANNEALING TEMP. FAHR. 


r. 


BY ANNEALING. 


I 


57,133 


1,037° 


56,678 


.025 


3 


53.774 


1, 111° 


52,186 


.029 


6 


53,185 


1,159° . 


46,212 


•131 


9 


52,040 


1,237° 


44,165 


.151 


12 


48,407 


Bright welding heat. 


39,333 


.187 


15 


48,407 


( ( i< < ( 


38,676 


.201 


18 


76,986 


(« < ( (( 


50,074 


•349 



T is the ultimate tensile resistance in pounds per square 

inch at ordinary temperatures, before annealing. 
T' is the same after annealing and cooling. 



The means of sets of five, three and four specimens of wire 
0.156 inch in diameter, exactly similarly treated, were, respect- 
ively : 



T — 89,162. 



T' = 48,144- 



r/ =: 50,889. 



The process of annealing is thus seen to decrease the ulti- 
mate tensile resistance a very considerable amount. In many 
cases, however, this may make the iron very much more valu- 
able, since annealing renders it much more ductile. If a struct- 
ure or machine is subject to shocks or sudden applications of 



Art. 32.] HARDENING AND TEMPERATURE. 247 

loading, a very stiff, hard iron, originally utterly unfit for the 

purpose, after being annealed might be used in its construc- 
tion with safety. 



Effect of Hardening on the Te7isile Resistance of Iron and 

Steel. 

It has been seen that annealing reduces the ultimate resist- 
ance of wrought iron. Experiments have shown that harden- 
ing, on the other hand, increases the resistance of both iron 
and steel, provided the hardening is done in a proper manner. 
If the hardening is accomplished by heating and sudden cool- 
ing in water, without subsequent tempering, the resistance of 
hard steel is very much diminished. This is probably due to 
the internal stresses induced by the sudden cooling. 

Knut Styffe (" Iron and Steel ") concluded from his experi- 
ments that " by heating and sudden cooling (hardening), the 
limit of elasticity is raised while the extensibility is diminished, 
not only in steel but also in iron." This results of the experi- 
ments by David Kirkaldy will be given hereafter. 



Variation of Te7isile Resistance with Increase of Temperature, 

Table XVI. has again been condensed from a similar one 
given in the Report of the Franklin Institute Committee. 

The third column gives the temperatures at which the ulti- 
mate tensile resistances in the fourth column were observed. 

The committee observed that the resistance of many irons 
increased with the temperature, to nearly the boiling point of 
mercury in some cases, while others remained unchanged until 
a temperature of 572° was reached. Above this point, how- 
ever, as a rule, they found the decrease of resistance, below the 
greatest, to vary about as the 2.6 power of {Temp. — 80°). 



248 



WROUGHT IRON IN TENSION 



[Art. 32. 



TABLE XVI. 





ULTIMATE TENSILE RESISTANCE 


TEMP, FAHR. 


ULTIMATE TENSILE RESISTANCE 




AT ORDINARY TEMPERATURE. 




AT OBSERVED TEMPERATURE. 


I 


56,736 


212 


67,939 


5 


62,646 


394 


67,765 


9 


49,782 


440 


59,085 


13 


52,542 


552 


55,939 


17 


53,385 


562 


59,623 


21 


66,724 


572 


66,620 


25 


76,071 


574 


65,387 


29 


59.234 


576 


66,065 


33 


45,757 


578 


53,465 


37 


59,530 


630 


60,010 


41 


52,542 


732 


53,378 


45 


59,219 


819 


55,892 


49 


59,219 


1,022 


37,410 


53 


54,768 


1,142 


18,672 


56 


53,426 


1,187 


21,910 


59 


54,758 


1,317 


18,913 



In the London " Engineering " of 30th July, 1880, is given 
a synopsis of some German experiments by Herr Kollmann, 
which is reproduced in Table XVII. The resistance of the 
materials at 0° Cent., or 32" Fahr., is taken as 100, and that at 
other temperatures as the proper proportional part of that 
number. 

It will be noticed that these German experiments show a 
much earlier decrease of resistance than those of the Franklin 
Institute. 

The results of some tests of a grade of charcoal boiler plate 
at three different temperatures are given in " Effect of low and 
high temperatures on steels 

Some French experiments by M. Baudrimont are given in 
the " Journal of the Franklin Institute " for 1850, by which he 
found that at the temperatures 32°, 212° and 392° Fahr., iron 



Art. 32.] 



EFFECT OF TEMPERATURE. 



249 



TABLE XVII. 



TEMPERATURE. 


FIBROUS IRON. 


FINE GRAINED IRON. 


BESSEMER STEEL. 






Cent. 


Fahr. 








0° 


32'' 


100 


100 


100 


100 


212 


100 


100 


100 


200 


392 


95 


100 


100 


300 


572 


90 


97 


94 


500 


932 


38 


44 


34 


700 


1,292 


16 


23 


18 


900 


1,652 


6 


12 


9 


1,000 


1,832 


4 


7 


7 



wire gave the following tensile resistances, in pounds per 
square inch, respectively : 

291,5 10.00 ; 271,602.00 ; 298,620.00 ; 

These resistances are most extraordinarily high, but, so far 
as the influence of variation of temperature is concerned, show 
nothing discordant with the preceding results. 

The same experimenter found the tensile resistances of 
gold, platinum, copper, silver and palladium to decrease, in 
every instance, as the temperature increased from 32° to 392° 
Fahr. 

In his " Useful Information for Engineers," Second Series, 
Sir Wm. Fairbairn gives the results of numerous experiments 
made on " short " specimens of plate and rivet iron at different 
temperatures. 



250 



WROUGHT ikON IN TENSION. 



[Art. 32. 



TABLE XVIII. 





BREAKING WEIGHT IN POUNDS PER 




TEMP., FAHR. 


SQUARE INCH. 


STRESS IN REFERENCE TO FIBRE. 


0° 


49.009 


With. 


60 


40,357 


Across. 


60 


43,406 


Across. 


60 


50,219 


With. 


1 10 


44,160 


Across. 


112 


42,088 


With. 


120 


40,625 


With. 


212 


39.935 


With. 


212 


45,680 


Across. 


212 


49>5oo 


With. 


270 


44,C20 


With. 


340 


49,968 


With. 


340 


42,088 


Across. 


395 


46,086 


With. 


Scarcely red. 


38,032 


Across. 


Dull red. 


30,513 


Across. 



In Table XVIII. will be found the results of his experi- 
ments on plate iron. On the whole, the table would seem to 
show a point of greatest resistance at about 270° to 300°, 
though so many irregularities exist that little or no law can be 
observed. In other words little or no decrease takes place at 
395° or below. Much diminution, however, is seen at " scarcely 
red " and more at '* dull red." 

Table XIX. shows the results of Fairbairn's experiments on 
rivet iron at different temperatures. The irregularities are less 
than those seen in Table XVIII., and a maximum would seem 
to exist at about 325°. 

The areas of the normal sections of the plate specimens 
varied from 0.6 to 0.8 square inch, while the sectional areas of 
the rivet-iron specimens were about 0.2 or 0.25 square inch. 

Other results for wrought iron will be found in Table IX. 
of Art. 35. 



Art. 32.] 



EFFECT OF TEMPERATURE. 



251 



TABLE XIX 



TEMPERATURE, 


BREAKING WEIGHT IN POUNDS 


TEMPERATURE, 


BREAKING WEIGHT IN POUNDS 


FAHR. 


PER SQUARE INCH. 


FAHR. 


PER SQUARE INCH. 


-30° 


63,239 


250 


82,174 


+ 60 


61,971 


270 


83,098 


66 


63,661 


310 


80,570 


114 


70,845 


325 


87,522 


212 


82,676 


415 


81,830 


212 


7-^,153 


435 


86,056 


212 


80,985 


Red heat. 


36,076 



All the preceding results, while irregular to some extent, 
show conclusively that no essential decrease in the tensile re- 
sistance of wrought iron takes place below about 500° FaJir., 
while a possible increase at that temperature may exist over 
that at any below, but that at about 1,000° it may lose more 
than a half of its resistance. These conclusions are of the 
greatest importance in the construction of boilers. 



Effect of Low Temperatures on Wro2ight Iron. 



It is a matter of common observation that many articles, 
large and small, are much more easily broken in very cold 
weather than at higher temiperatures. These breakages are 
undoubtedly frequently due to the circumstances in which the 
piece broken is found at the time of failure, either partly or 
wholly. 

The frozen, and consequently less yielding, condition of the 



252 WROUGHT IRON IN TENSION. [Art. 32. 

ground in the winter is unquestionably a very potent factor in 
failures or tires and axles of railway rolling stock, but it is at 
least an open question whether it is the sole cause. 

A number of investigators have made numerous experi- 
ments with the object of determining the effect of low tem- 
peratures on the resistance of wrought iron in different forms. 

From the results of these experiments, however, they have 
drawn the most discordant conclusions. In some cases this 
arises from the fact that the tests have not been made under 
the same circumstances, or Jiave not been of the same kind. 

Knut Styffe ('' Iron and Steel ") made the following '' Re- 
sume of Results of Experiments on Tension at different Tem- 
peratures : " 

I. '' That the absolute strength of iron and steel is not dimin- 
. ished by cold, but that even at the lowest temperature 
which ever occurs in Sweden it is at least as great as 
at the ordinary temperature (about 60° fahr.). . . . 

3. " That neither in steel nor in iron is the extensibility less in 

severe cold than at the ordinary temperature ; . . . 

4. '^ That the limit of elasticity in both steel and iron lies higher 

in severe cold ; . . ." 

He concluded from his experiments that the common im- 
pression of increased weakness and brittleness with a low de- 
gree of temperature is entirely erroneous. His tests, however, 
were wholly with tension gradually applied, and could support 
no conclusion in regard to other conditions. 

The translator of Styffe's work, Christer P. Sandberg, made 
some experiments in order to determine the effect of shocks at 
different temperatures, i.e.^ ordinary and low. These were also 
made in Sweden, and by dropping heavy weights, from differ- 
ent heights, on rails supported at each extremity. The records 
of these tests may be found in the translator's Appendix to 
Styffe's work. 



Art. 32.] EFFECT OF TEMPERATURE. 253 

The following are Sandberg's conclusions, and they will be 
observed to be directly opposed to those of Styffe : 

1. '' That for such iron as is usually employed for rails in the 

three principal rail-making countries (Wales, France 
and Belgium), the breaking strain, as tested by sudden 
blows or shocks, is considerably influenced by cold ; 
such iron exhibiting at 10° Fahr. only one-third to 
one-fourth of the strength which it possess at 84° 
Fahr, 

2. " That the ductility and flexibility of such iron is also much 

affected by cold; rails broken at 10° Fahr. showing 
on an average a permanent deflection of less than one 
inch, whilst the other halves of the same rails, broken 
at 84° Fahr.f showed a set of more than four inches 
before fracture. 

3. " That at summer heat the strength of the Aberdare rails 

was 20 per cent, greater than that of the Creusot rails ; 
but that in winter the latter were 30 per cent, stronger 
than the former." 

All these experiments were made previous to 1869, and 
with iron rails. -^ 

Prof. Thurston, from his own experiments and those of 
others, concludes (Trans. Am. Soc. of Civ. Engrs., Vol. III., p. 
30), " That with good materials, cold does not produce injury, 
but actually improves their power of resisting stress and in- 
creases their resilience. 

" That the influence of impurities, of various methods of 
manufacture, of changes of density with temperature, and of 
the causes which produce a concentration of the action of 
rapidly produced distortion and of quick blows, are subjects 
which still require careful investigation." 

He considers it probable that the cold-shortening effect of 
phosphorus is intensified at low temperatures. 



254 WROUGHT IRON IN TENSION. [Art. 32. 

After observing the failures on the railroads coming under 
their observation, the Railroad Commissioners of Massachu- 
setts reported in 1874 that, in their opinion, neither iron nor 
steel attained any greater degree of brittleness, or became any- 
more *' unreliable for mechanical purposes " at low tempera- 
tures than at ordinary. They did not observe as a " rule that 
the most breakages " occurred " on the coldest days." 

They further stated that *' the introduction of steel in place 
of iron rails, has caused an almost complete cessation of the 
breakage of rails." 

Thus it is seen that the subject is most thoroughly involved 
in confusion. It seems, however, to be established that the 
resistance of iron, at a low temperature, to a steady strain, is 
not diminished, while it may, perhaps, be increased. 

Its resistance to shocks, at low temperatures, is probably 
very much affected by its quality, mode of manufacture or 
chemical composition, and these should always be taken into 
consideration when experiments are made. 

The Report of the Mass. Railroad Commissioners would 
indicate that steel rails resist shocks at low temperatures better 
than iron ones. 

# 

Iron Wire. 

Mr. John A. Roebling found by his tests that the English 
wire used in the Niagara Falls Suspension Bridge gave an ulti- 
mate tensile resistance of about 98,500.00 pounds per square 
inch C Papers and Practice Illustrative of Public Works." John 
Weale, London, 1856). This wire was about 0.145 inch in 
diameter. 

The Committee of the Franklin Institute made thirteen 
tests of some iron wire one-third of an inch in diameter, of 
which the highest, lowest and mean ultimate resistances in 
pounds per square inch of original section were as follows : 



Art. 32.] 



IRON WIRE. 



255 



Highest 88,354.00 pounds. 

Mean 84, 186.00 pounds. 

Lowest 72,325.00 pounds. 

The results of other tests by the same committee have 
already been given under " Effect of annealing r 

TABLE XX. 





ULTIMATE TENSILE RESISTANCE IN POUNDS PER 


CONTRACTION OF 


ORIGINAL DIAMETER 


SQUARE INCH OF 


ORIGINAL AREA 








OF SECTION. 




Original Area. 


Fractured Area. 




0.122 


94,871 


179,032 


0.47 


0. 123 


87,395 


162,500 


0.462 


0.124 


89,256 


145,946 


0.388 


0.125 


88,618 


137,974 


0.358 


0.122 


92,308 


168,750 


0.453 


0.124 


91,735 


156,338 


0.413 


0.124 


90,082 


170,313 


0.471 


0.122 


92,308 


168,750 


0.453 


0. 124 


91,735 


173,437 


0.471 


0.124 


86,776 


164,063 


0.471 


0.125 


87,805 


156,522 


0.439 


0. 124 


86,776 


152,174 


0.43 



Table XX. is a condensed form of one given in the "' Trans- 
actions of the Am. Soc. Civ. Engrs.," Vol. III., p. 212, and con- 
tains an account of the tests made by Prof. R. H. Thurston on 
some wires that had been in use 32 years in the cables of the 
Fairmount Suspension Bridge at Philadelphia. It is both in- 
teresting and important to observe that the long service can- 
not have appreciably injured either the ducility or ultimate 
resistance of the wire. 

Table XXI. contains the records of tests on other wire, at 
the same time (1875), by Prof. Thurston. The small reduction 



256 



WROUGHT IRON IN TENSION. 



[Art. 32. 



of diameter at fracture shows the iron to have been not very 
ductile. It will also be noticed that the smaller diameters give 
much the highest resistances. 

TABLE XXI. 







ULTIMATE RESISTANCE IN POUNDS PER 


ORIGINAL DIAMETER. 


DIAMETER AFTER FRACTURE. 


SQUARE INCH OF ORIGINAL AREA. 


0.134 


0.133 


92,890 


0.1205 


O.II85 


84,442 


0.08 


0.0795 • 


94,299 


0.071 


0.068 


90,384 


0.0535 


0.0532 


105,871 


0.029 


0.029 


113,546 



According to Weisbach ('' Mechanics of Engineering, etc.,'* 
Vol. I., 4*'' Edit.), Lagerhjelm and Brix found the mean value 
of the ultimate resistance, for a large number of tests of 
wrought-iron v/ire with diameters varying from 0.0833 to 0.125 
inch, to be 98,000.00 pounds per square inch for unannealed 
wire, and 64,500.00 pounds for annealed. 

Morin, in his " Mecanique Practique," gives the following 
for unannealed iron wire, after changing his results to pounds 
per square inch : 

Mean for diameters of 0.039 to 0.118 inch. . . . 85,000.00 (nearly). 

Highest for diameters of 0.02 to 0.039 inch. . . 114,000.00 (nearly). 

Lowest for large diameter 71,000.00 (nearly). 

For a special grade (" I'Aigle ") 128,000.00 (nearly). 

SirWm. Fairbairn (''Useful information for Engineers, 3d 
Series," p. 282) gives the following as the results of experi- 



Art. 32.] 



IRON WIRE. 



257 



ments on various kinds of English iron wire. These experi- 
ments resulted from investigations relating to the fabrication 
of a submarine Atlantic cable. 



KIND OF WIRE. 



Haematite 

Homogeneous 

Special Homogeneous 

Charcoal 

Galvanized 

Homogeneous 

Homogeneous 

Charcoal 

Homogeneous 

Charcoal 

Haematite, S. 3 

Haematite, S. 4 

Homogeneous 

Homogeneous 

Homogeneous 

Homogeneous 

Special Charcoal 



DIAMETER. 


ULT. RESIST. 


Inch. 


Pounds. 


0.087 


109,300 


0.095 


134,000 


0.097 


115,000 


0.093 


110,400 


0.098 


86,200 


0.089 


104,500 


0.091 


192,200 


0.091 


92,200 


0.088 


106,900 


0.093 


80,960 


0.089 


88,400 


0.095 


105,800 


0.180 


45,200 


0. 148 


61,050 


0.095 


134,000 


0.095 


77,600 


0.095 


105,800 



o 



Inch. 
0.280 
366 
0.267 
0.173 
0.198 
o. 190 

0.712 
0.198 
0.218 
0.320 

o. 171 
0.366 
0.480 

0.550 
0.346 
o. 116 
0.170 



The ultimate resistance is in pounds per square inch, and 
the stretch is the total amount for 50 inches of length. 
Reviewing the values given, it appears : 

1. TJiat wire is the strongest form in which iron can be used to 

resist tensile stress ; 

2. That J as a rule, the ultimate tensile resistance increases as the 

diameter of the wire decreases. 



Tensile Resistance of Shape Iron, 

The phenomena exhibited in the fracture of shape iron de- 
pend, to a great extent, on the character of the piles from 
which it is rolled. The webs of Es and Is are sometimes rolled 
from old rails in connection with double refined iron in the 
17 



258 WROUGHT IRON IN TENSION. [Art. 32. 

flanges. In such cases, specimens cut from the web will fre- 
quently, if not usually, show a high intensity of ultimate resist- 
ance, but very little ductility, while those cut from the flanges 
will give good records of both kinds. 

In general, shapes will offer less tensile resistance than 
either bars or rods, yet small specimens cut from good shape 
iron will give values ranging from 52,000 to 58,000 pounds per 
square inch, with ductility little less than that of Qs and Qs, 

English Wrought Iron, 

A great number of experiments on English wrought iron 
have been made by Sir Wm. Fairbairn, David Kirkaldy, and 
others. A record of Fairbairn's experiments may be found in 
his " Useful Information for Engineers," while an account of 
those of the latter is given in *' Experiments on Wrought Iron 
and Steel," by David Kirkaldy, Glasgow, 1863. 

B. B. Stoney, in his ^* Theory of Strains in Girders and Simi- 
lar Structures," summarizes Kirkaldy's results, in pounds per 
square inch, as follows : 

Mean of 188 rolled bars 57,555-00 

Mean of 72 angle irons and straps 54,729.00 

Mean of 167 plates, lengthwise 50,737.00 

Mean of 160 plates, crosswise 46,171.00 

It should be stated that these means include some Russian 
and Swedish irons, also that the bars were small ones. 

These results do not differ much from quantities for cor- 
responding grades of American iron. 

Fracture of Wrought Iron. 

The characteristic fracture of wrought Iron broken in ten- 
sion, either directly or transversely, is rather coarsely fibrous, 



Art. 32.] FRACTURE AND CRYSTALLIZATION. 259 

not unfrequently exhibiting a few bright granular spots which, 
in rare cases, may possibly be crystalline. This characteristic 
(fibrous) fracture is always produced by the steady application 
of an external force, under the influence of which the piece is 
drawn out in jagged points at the place of failure. 

The best of fibrous wrought iron, however, will exhibit a 
granular fracture if broken suddenly. In making tests, there- 
fore, it is of the greatest importance to observe and direct the 
mode of application of the external forces producing fracture. 

When some grades of iron in bars are broken transversely 
by shocks (such as are produced by falling weights), a phenome- 
non known as " barking " is produced. A skin of metal from 
a sixteenth to an eighth of an inch in thickness, on the tension 
side of the bent piece, tears apart and separates from the core 
of the bar. At the place of fracture and on each side of it, 
this skin or "■ bark " remains essentially straight. This kind of 
fracture shows remarkably well the fibrous character of wrought 
iron ; it is simply the separation of the fibres near the outside 
of the bar from those within. 

Crystallization of Wrought Iron, 

The subject of crystallization of wrought iron is one about 
which there is much dispute. In '* Strength of Wrought Iron 
and Chain Cables," by Beardslee, as abridged by Kent, p. 36, 
the following is given as the opinion or view of the United 
States Testing Commission : *^ The question as to whether 
crystallization can be produced in iron by stress, or by repeti- 
tion of stress with alternations of rest, or by vibration, has 
been much discussed, and very opposite views are entertained 
by experts. 

*' We have met with but one unmistakable instance of crystal- 
lization which was probably produced by alternations of severe 
stress, sudden strains, recoils and rest." 

The committee then state the case of a connecting-rod, 



26o WROUGHT IRON IiV TENSION. [Art. 32. 

carefully made of the best quality of wrought-iron scrap, which 
had been used in a testing machine for forty years, in the 
Navy Yard at Washington. It was five inches in diameter, 
but one day, while in use it suddenly broke under a stress 
(total) of less than 200,000 pounds. *' The surface of the fract- 
ured ends showed well-defined crystallization, the facets being 
large and bright as mica." 

The data at hand, at present, are not sufficient for a decision 
of the question, but it may be confidently stated that in many 
cases granulation has been mistaken for crystallization. 



Elevation of Ultimate Resistance and Elastic Limit, 

It was first observed by Prof. R. H. Thurston and Com- 
mander L. A. Beardslee, U. S. N., independently, in this coun- 
try, that if wrought iron be subjected to a stress beyond its 
elastic limit, but not beyond its ultimate resistance, and then 
allowed to " rest " for a definite interval of time, a considerable 
increase of elastic limit and ulti7nate resistance may be expe- 
rienced. In other words, the application of stress and subse- 
quent " rest " increases the resistance of wrought iron. 

This *' rest " may be an entire release from stress or a sim- 
ple holding the test piece at a given intensity. 

Prof. Thurston's investigations were on torsion, while those 
of the United States Commission were on tension, and will be 
given here. 

The Commission prepared twelve specimens and subjected 
them to an intensity of stress equal to the ultimate resistance 
of the material, without breaking the specimens. These were 
then allowed to rest, entirely free from stress, from twenty-four 
to thirty hours, after which period they were again stressed 
until broken. 

The gain in ultimate resistance by the rest was found to 
vary from 4.4 to 17 per cent. 



Art. 32.] ELEVATION OF RESISTANCE. 26 1 

These tests, remark the committee, seem to indicate that 
the tough fibrous irons gained the most, while those which 
broke with a steeUike fracture gained the least. 

Before the rest, the stress which produced the first perma- 
nent elongation was about 65 per cent, of the ultimate resist- 
ance, but after the rest the two were nearly identical. 

The committee then took forty-two other specimens and 
subjected them to precisely the same operations, except that 
the rest periods varied from one minute to six months. 

The gains were as follows : 

In less than I hour 1. 1 per cent., mean of 5 tests. 

In less than 8 and over i hour 3.8 per cent., mean of 8 tests. 

In 3 days 16.2 per cent., mean of 10 tests. 

In 8 days 1 7. 8 per cent. , mean of 2 tests. 

Between 8 and 43 days 15.3 per cent. , mean of 5 tests. 

In 6 months 17.9 per cent., mean of 12 tests. 

After seven other experiments involving a rest of 24 hours, 
with an average gain of 15.4 per cent., the committee con- 
cluded *' that at the end of one day the result is, with very 
ductile irons, practically accomplished." 

The manifestation of this phenomenon in different grades 
of iron was then investigated. 

*' Thirteen pieces were prepared, five of which were of soft 
charcoal bloom boiler iron, five of coarse contract chain iron, 
and three of a fine-grained bar of . . . very pure iron with 
high tenacity." 

After testing these specimens subsequent to an eighteen 
hours' rest, the committee state (Kent's abridgment) : 

"These experiments confirmed the opinion already formed, 
and indicate that a bridge, cable, or other structure, composed 
of iron of either of the latter two varieties, will receive com- 
paratively slight benefit from the operation of this law ; while 
ductile fibrous metal . . . gains . . . to a great extent 
by the effect of strains already withstood." The gain in these 



262 WROUGHT IRON IN TENSION. [Art. 32. 

specimens varied from about 3 per cent, (for the coarse iron) 
to about 18 per cent, (for the soft iron). 

Again, two sets of specimens were prepared : one from the 
two portions of fractured bars after having been pulled asunder, 
the other from the bars in their normal condition. After a rest 
of several days the first set showed a gain over the second in 
ultimate resistance, varying from about 8 to 39 per cent., the 
higher values belonging to the more ductile irons. 

Bauschinger s Experiments on the CJiange of Elastic Limit and 

Coefficient of Elasticity. 

In " Der Civilingenieur," Heft 5, for 1881, are contained the 
results of the experiments of Prof. Bauschinger, of Munich. 
The observations in these experiments were made by the aid 
of a piece of apparatus which gave the elongations (all experi- 
ments were tensile) in ten-millionths of a metre, or approxi- 
mately in joVo-o" of an inch. An extraordinarily high degree 
of accuracy was therefore attained. 

Prof. Bauschinger's elastic limit was strictly a proportion- 
ality limit between stresses and strains. He also observed 
what may be called the ''stretch-limit " (Ger., Streckgrenze), at 
which point the stretching or elongation suddenly increases and 
continues to increase for more than a minute after the appli- 
cation of the stress. In ordinary experimenting this point has 
probably frequently been considered the elastic limit. 

The test pieces were subjected to loads which gradually 
increased from zero by an increment a little less than 3,000 
pounds per square inch, each load having been allowed to act 
one minute before adding the succeeding increment. At inter- 
vals of the loading separated by about 11,500 or 12,000 pounds 
per square inch, each piece was entirely unloaded and allowed 
to remain so for 15 or 20 minutes. After the "stretch-limit" 
was found the piece was subjected to a final load somewhat 
greater than the '' stretch-limit," and then entirely unloaded. 



Art. 32.] 



BA USCHINGER'S EXPERIMENTS. 



263 



In some cases the piece was immediately put through the 
same process of testing either once or a number of times, and 
the results of such tests will be found in the columns of the 
following tables, indicated by the contraction "■ IinyT 

In the remaining cases intervals of time, shown at the tops 
of the columns, were allowed to elapse between any one test 
and the succeeding ®ne. " .^^ - 

The tables, Nos. I to 7 inclusive, give the results of the 
experiments on seven specimens of a grade of iron called 
" Schweisseisen ** (weld iron). These specimens were a very 
little less than i inch (25 millimetres) in diameter. Nos. I and 
2 were about 32 inches long, and the others about 16 inches 
long. 

•Tables No. 8 to 13, inclusive, give the results obtained with 
Krupps " Flusseisen." These specimens were about one inch 
in diameter and sixteen inches long. 

The tables have been condensed from those given by Bau- 
schinger and reduced to English measures. 

The following is the notation : 

E, L. = elastic limit in pounds per square inch. 
S.-L. = stretch limit in pounds per square inch. 

F. L. = final load in pounds per square inch. 

E, = coefficient of elasticity in pounds per sq. in. 



IVeld Iron. 





IN ORIGINAL CONDI- 








KO. I. 


TION. 


im'y. 


im'y. 


im'y. 


E. L. 


20,110 


14,370 


14,900 


15,500 


S.-L. 


27,300 


31,600 


41,700 


49,500 


F. L. 


31,600 


40, 200 


47,700 






E. 


39,293,000 


27,928,000 


27,672,000 


27,544,000 



264 



WROUGHT IRON IN TENSION. [Art. 32. 



Weld Iron. 





IN ORIGINAL CONDI- 








NO. 2. 


TION. 


AFTER 19 HRS. 


AFTER 27 HRS. 


AFTER 24 HRS. 


E. L. 


20,110 


28,970 


35,500 


39,100 


S.-L. 


28,700 


34,750 


44,300 


45,100 


F. L. 


31,600 


40,540 


47,300 






E. 


29,037,000 


28,923,000 


28,198,000 


28,241,000 



Weld Iron. 





IN ORIGINAL CONDI- 








KO. 3. 


TION. 


AFTER 51 HRS. 


AFTER 41 HRS. 


AFTER 45 HRS. 


E. L. 


23,164 


28,440 


39,080 


45,580 


S.-L. 


29,000 


34,750 


45,090 






F. L. 


31,900 


40,540 


48,100 





E. 


29,208,000 


28,397,000 


28,483,000 


28,170,000 



Weld Iron. 





IN ORIGINAL CONDI- 








NO. 4. 


TION. 


AFTER 80 HRS. 


AFTER 68 HRS. 


AFTER 64 HRS. 


E. L. 


22,890 


31,900 


35,340 


43,800 


S.-L. 


30,050 


34,750 


44,170 






F. L. 


31,470 


40,540 


47,110 






E. 


29,293,000 


28,810,000 


28,227,000 


28,696,000 



Art. 32.] BAUSCHINGER'S EXPERIMENTS. 



265 



Weld Iron. 


NO. 5. 


IN ORIGINAL CONDI- 
TION. 


im'y. 


AFTER 63 HRS. 


im'y. 


E. L 
S.-L. 

F. L. 
E. 


21,070 

30,670 

34,611 

29,293,000 


14,720 
35.320 
42,700 

28,312,000 


42,090 

48,110 

51,110 

28,056,000 


15,260 
51,860 


26,705,000 


Weld Iron. 


NO. 6. 


IN ORIGINAL CONDI- 
TION. 


AFTER 48.5 HRS. 


AFTER 44.5 HRS. 


AFTER 49 HRS. 


E. L. 

5^ -I. 


27,730 

32,120 

35.040 

29,720,000 


26,720 

37,xio 

43,040 
28,639,000 


33,350 

45,480 

51,550 

28,483,000 


24,940 


F. L. 
E. 




28,881,000 


Weld Iroft. 


NO. 7. 


IN ORIGINAL CONDI- 
TION. 


AFTER 47 HRS. 


AFTER 50.5 HRS. 


AFTER 42.5 HRS. 


E. L. 
S.-L 


20,110 

30,160 

34,470 

28,668,000 


26,720 

38,590 

43,040 

28,611,000 


27,170 

45,290 

51,320 

28,568,000 


18,540 


F. L 




E. 


29,592,000 



266 



WROUGHT IRON IN TENSION. 



[Art. 32. 



Melted Wrought Iron. 



NO. 8. 


IN ORIGINAL CONDI- 
TION. 


im'y. 


im'y. 


im'y. 


E. L. 
S.-L. 

F. L. 
E. 


35,340 

36,750 

45,230 

31,256,000 




8,990 

53,890 

59,8So 

31,483,000 


9,230 

58,490 


46,910 
52,770 


30,488,000 





Melted Wrought Iron. 





IN ORIGINAL CONDI- 








NO. 9. 


TION. 


im'y. 


im'y. 


im'y. 


E. L. 


37,870 


5,770 


14,720 


15,160 


S.-L. 


42,080 


46,140 


53,000 


60,630 


F. L. 


44,880 


51,920 


58,880 






E. 


32,379,000 


31,796,000 


29,947,000 


28,213,000 



Melted Wrought Iron. 





IN ORIGINAL CONDI- 








NO. 10. 


TION. 


AFTER 3 HRS. 


AFTER 15 HRS. 


AFTER 7 HRS. 


E. L. 


33,790 


11,490 


17,730 


15,260 


S.-L. 


36,600 


43,090 


53,210 


61,020 


F. L. 


42,230 


51,704 


59,130 






E. 


31,881,000 


31,953,000 


31,895,000 


32,393,000 



Art. 32.] BAUSCHINGER'S EXPERIMENTS. 



267 



Melted Wrought Iron. 





IN ORIGINAL CONDI- 








NO. 11. 


TION. 


AFTER 2.5 HRS. 


AFTER 15.5 HRS. 


AFTER 5.5 HRS. 


E. L. 


33,930 


11,590 


11,774 


12,070 


S.-L. 


39,570 


43,442 


53,000 


60,380 


F. L. 


42,404 


52,130 


58,880 






E. 


32,536,000 


32,237,000 


32,222,000 


31,085,000 



Melted Wrought Iron. 





IN ORIGINAL CONDI- 








NO. 12. 


TION. 


AFTER 51 HRS. 


AFTER 47 HRS. 


AFTER 46 HRS. 


E. L. 


36,900 


43,800 


39.820 


40,950 


S.-L. 


39,731 


49,640 


55,940 


60,630 


F. L. 


42,630 


52,560 


58,880 






E. 


32,479,000 


31,754,000 


31,696,000 


31,568,000 



Melted Wrought Iron. 



NO. 13. 


IN ORIGINAL 
CONDITION. 


AFTER 
43.5 HRS. 


AFTER 

54 HRS. 


AFTER 
44.5 HRS, 


AFTER 
45.5 HRS. 


AFTER 
10 DAYS. 


E. L. 
S.-L. 

F. L. 
E. 




35,340 

36,745 

42,400 

32,165,000 


38,930 

47,600 

51,920 

31,298,000 


41,740 

56,640 

59,630 

31,454,000 


42,720 


61,560 










31,853,000 


31,440,000 


32,364,000 



268 WROUGHT IRON IN TENSION. [Art. 32. 

During the progress of the various tests, the bars Nos. 6, 7, 
9, II and 12 were subjected to shocks in addition to the static 
tests. These shocks were produced by striking the test piece 
on its end by a hammer. It does not appear that these blows 
of the hammer perceptibly influenced the results. 

The ultimate resistance of the weld iron was found to vary 
from 55,300 to 58,870 pounds per square inch. That of the 
melted wrought iron was about 65,000 pounds per square inch. 

Although there are some irregularities, the following gen- 
eral conclusions may be drawn from the tables: 

By " immediate " testing the elastic limit of weld iron is 
very much decreased. 

With a rest (entirely free from load) between the tests, the 
elastic limit of weld iron is very much increased. 

The greatest proportional gain, except in the case of previ- 
ous immediate testing, seems to be acquired after a rest no 
greater than twenty hours. 

Bar No. 6 is seen to give anomalous results. 

In all cases of the weld iron the stretch-limit is considerably 
raised by repeated testing. 

In no case is the coefficient of elasticity, after once testing, 
equal to its original value ; as a rule, a steady decrease is seen 
to take place by repeated testing, but there are some ex- 
ceptions. 

The elastic limit of '' Flusseisen," after repeated testing, is 
found to be much less than its original value until the length 
of rest becomes about fifty hours. 

The stretch-limit of the same metal is invariably raised by 
repeated testing, either with or without " rests." 

In nearly all the cases of Nos. 8 to 13, the coefficient of 
elasticity is found to be slightly decreased by repeated testing. 

For a very clear and detailed account of these experiments 
reference must be made to the ** Civilingenieur." 



Art. 32.] SUDDEN STRESS. 269 



Resistance of Bar Iron to Suddenly Applied Stress. 

If tensile stress is suddenly applied to a bar of wrought iron, 
both its ultimate resistance and elongation will be very materi- 
ally decreased. 

As a mean of a number of tests, Mr. David Kirkaldy 
(" Experiments on Wrought Iron and Steel ") found with sud- 
denly applied stress an ultimate resistance of 46,500 pounds 
per square inch, while with stress gradually applied it rose to 
57,200 pounds. 

In the former case the elongation was about 20 per cent., 
and as high as 24.6 per cent, in the latter. 

It is thus seen that the mode of application of external 
force not only affects the character of the fracture of the iron, 
but also its ultimate resistance and elongation. 

It will hereafter be seen that similar observations apply to 
other metals than wrought iron. 

Reduction of Resistance Between the Ultimate and Breaking 

Point. 

It has already been observed that the ultimate tensile re- 
sistance of wrought iron is the greatest tensile resistance which 
it offers to being pulled asunder, and that a test specimen 
finally parts at much less than the ultimate resistance. This is 
due to the ductility of the iron, which allows it to " pull out " 
or stretch, thus decreasing the cross section as well as the 
actual resisting capacity of the metal. 

The ultimate resistance, therefore^ is not exerted on the final 
section of fracture, but on a section somewhat greater ; referring 
it (the ultimate resistance) to the section of fracture, then, may 
mean little or nothing. 

The United States Commission made six tests, for the pur- 
pose of determining this reduction, on some specimens which 



270 WROUGHT IRON IN TENSION. [Art. 32. 

had previously been stressed with a subsequent rest. The 
highest, lowest, and mean losses were as follows : 

Highest 14 . 5 per cent. 

Mean 13.8 per cent. 

Lowest 1 2 . 9 per cent. 

It was observed from a number of specimens, by the same 
commission, that the reduction of area at the instant of ulti- 
mate resistance (or greatest resistance) was about one-half, and 
the elongation or strain a little over three-quarters, of the cor- 
responding quantities at the instant of fracture, supposing 
failure to be produced by a steady strain. 

Some further observations seemed to show that if failure 
were produced by shock, the final contraction would be nearly 
the same as at the instant of greatest resistance in the case of 
a steady failure. 

Effects of Chemical Constitution. 

While It is well known that the resistance of wrought iron 
to tension varies greatly with the chemical composition, it is 
yet uncertain just what influence most of the foreign elements, 
found in iron exert, either individually or collectively. This 
will be apparent on examining Table XXII., taken from the 
report of the three committees of the United States Commis- 
sion, to which allusion has here been so frequently made 
before. 

The first part of the table represents the relative values of 
sixteen different irons in reference to their physical character- 
istics, one being the highest. The second part shows the 
amount of the various elements named in the left-hand lower 
column, found in the corresponding irons, /. ^., each vertical 
column belongs to one iron. 

An inspection of the table will make very evident the diffi- 



Art. 32.] 



CHEMICAL CONSTITUTION. 



271 



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2/2 WROUGHT IRON IN TENSION. [Art. 32. 

culty of drawing definite conclusions in regard to any one 
element. 

For a detailed discussion of these results reference must be 
made to the report. 

Kirkaldy s Conclusions. 

The following conclusions were deduced by Mr. Kirkaldy 
from the results of his experiments. As will be seen, they 
belong to both wrought iron and steel in tension, and are 
taken from his "■ Experiments on Wrought Iron and Steel," 
1861 : 

1. The breaking strain does not indicate the quality, as hitherto assumed. 

2. A high breaking strain may be due to the iron being of superior quality, 
dense, fine, and moderately soft, or simply to its being very hard and unyielding. 

3. A lorv breaking strain may be due to looseness and coarseness in the texture, 
or to extreme softness, although very close and fine in quality. 

4. The contraction of area at fracture, previously overlooked, forms an essential 
element in estimating the quality of specimens. 

5. The respective merits of various specimens can be correctly ascertained by 
comparing the breaking strain jointly with the contraction of area. 

6. Inferior qualities show a much greater variation in the breaking strain than 
superior. 

7. Greater differences exist between small and large bars in coarse than in fine 
varieties. 

8. The prevailing opinion of a rough bar being stronger than a turned one is 
en-oneous. 

9. Rolled bars are slightly hardened by being forged down. 

10. The breaking strain and contraction of area of iron plates are greater in the 
direction in which they are rolled than in a transverse direction. 

11. A very slight difference exists between specimens from the centre and speci- 
mens from the outside of crank shafts. 

12. The breaking strain and contraction of area are greater in those specimens 
cut lengthways out of crank shafts than in those cut crossways. 

13. The breaking strain of steel, when taken alone, gives no clue to the real 
qualities of various kinds of that metal. 

14. The contraction of area at fracture of specimens of steel must be ascertained 
as well as in those of iron. 

15. The breaking %\.x2^x\, jointly w^ith the contraction of area, affords the means 
of comparing the peculiarities in various lots of specimens. 



Art. 32.] KIRKALDY'S CONCLUSIONS. 273 

16. Some descriptions of steel are found to be very hard, and, consequently, 
suitable for some purposes ; whilst others are extremely soft, and equally suitable 
for other uses. 

17. The breaking strain and contraction of area oi puddled steel plates, as in iron 
plates, are greater in the direction in which they are rolled ; whereas in cast steel 
they are less. 

18. Iron, when fractured suddenly, presents invariably a crystalline appearance ; 
when fractured slowly, its appearance is invariably fibrous. 

ig. The appearance may be changed from fibrous to crystalline by merely al- 
tering the shape of specimen, so as to render it more liable to snap. 

20. The appearance may be changed by varying the treatment, so as to render 
the iron harder and more liable to snap. 

21. The appearance may be changed by applying the strain so suddenly as to 
render the specimen more liable to snap, from having less time to stretch. 

22. Iron is less liable to snap the more it is worked and rolled. 

23. The " skin " or outer part of the iron is somewhat harder than the inner 
part, as shown by appearance of fracture in rough and turned bars. 

24. The mixed character of the scrap iron used in large forgings is proved by 
the singularly varied appearance of the fractures of specimens cut out of crank 
shafts. 

25. The texture of various kinds of wrought iron is beautifully developed by 
immersion in dilute hydrochloric acid, which, acting on the surrounding impurities, 
exposes the metallic portion alone for examination. 

26. In the fibrous fractures the threads are drawn out, and are viewed externally, 
whilst in the crystalline fractures the threads are snapped across in clusters, and are 
viewed internally or sectionally. In the latter cases the fracture of the specimen is 
always at right angles to the length ; in the former it is more or less irregular. 

27. Steel invariably presents, when fractured slowly, a silky fibrous appearance ; 
when fractured suddenly, the appearance is invariably granular, in which case also 
the fracture is always at right angles to the length ; when the fracture is fibrous, the 
angle diverges always more or less from 90°. 

28. The granular appearance presented by steel suddenly fractured is nearly 
free of lustre, and unlike the brilliant crystalline appearance of iron suddenly fract- 
ured ; the two combined in the same specimen are shown in iron bolts partly con- 
verted into steel. 

29. Steel which previously broke with a silky fibrous appearance, is changed 
into granular by being hardened. 

30. The little additional time required in testing those specimens, whose rate of 
elongation was noted, had no injurious effect in lessening the amount of breaking 
strain, as imagined by some. 

31. The rate of elongation varies not only extremely in different qualities, but 
also to a considerable extent in specimens of the same brand. 

32. The specimens were generally found to stretch equally throughout their 

18 



274 WROUGHT IRON IN TENSION. [Art. 32. 

length until close upon rupture, when they more or less suddenly drew out, usually 
at one part only, sometimes at two, and, in a few exceptional cases, at three differ- 
ent places. 

33. The ratio of ultimate elongation may be greater in short than in long bars 
in some descriptions of iron, whilst in others the ratio is not affected by difference 
in the length. 

34. The lateral dimensions of specimens forms an important element in com- 
paring either the rate of, or the ultimate, elongation — a circumstance which has 
been hitherto overlooked. 

35. Steel is reduced in strength by being hardened in water, while the strength 
is vastly increased by being hardened in oil. 

36. The higher steel is heated (without, of course, running the risk of being 
burned) the greater is the increase of strength by being plunged into oil. 

37. In a highly converted or hard steel the increase in strength and in hardness 
is greater than in a less converted or soft steel. 

38. Heated steel, by being plunged into oil instead of water is not only consid- 
erably hardened, but toughened by the treatment. 

39. Steel plates hardened in oil, and joined together with rivets, are fully equal 
in strength to an unjointed soft plate, or the loss of strength by riveting is more 
than counterbalanced by the increase in strength by hardening in oil. 

40. Steel rivets, fully larger in diameter than those used in riveting iron plates 
of the same thickness, being found to be greatly too small for riveting steel plates, 
the probability is suggested that the proper proportion for iron rivets is not, as 
generally assumed, a diameter equal to the thickness of the two plates to be joined. 

41. The shearing strain of steel rivets is found to be about a fourth less than 
the tensile strain. 

42. Iron bolts, case-hardened, bore a less breaking strain than when wholly 
iron, owing to the superior tenacity of the small proportion of steel being more than 
counterbalanced by the greater ductility of the remaining portion of iron. 

43. Iron highly heated and suddenly cooled in water is hardened, and the break- 
ing strain, when gradually applied, increased, but at the same time it is rendered 
more liable to snap. 

44. Iron, like steel, is softened, and the breaking strain reduced, by being 
heated and allowed to cool slowly. 

45. Iron subject to the cold-rolling process has its breaking strain greatly in- 
creased by being made extremely hard, and not by being ''consolidated," as pre- 
viously supposed. 

46. Specimens cut out of crank-shafts are improved by additional hammering. 

47. The galvanizing or tinning of iron plates produces no sensible effects on 
plates of the thickness experimented on. The result, however, may be different, 
should the plates be extremely thin. 

48. The breaking strain is materially affected by the shape of the specimen. 
Thus the amount borne was much less when the diameter was uniform for some 



Art. 32.] KIRKALDY'S CONCLUSIONS. 275 

inches of the length than when confined to a small portion — a peculiarity previously 
unascertained, and not even suspected. 

49. It is necessary to know correctly the exact conditions under which any 
tests are made before we can equitably compare results obtained from different 
quarters. 

50. The startling discrepancy between experiments made at the Royal Arsenal, 
and by the writer, is due to the difference in the shape of the respective specimens, 
and not to the difference in the two testing machines. 

51. In screwed bolts the breaking strain is found to be greater when old dies 
are used in their formation than when the dies are new, owing to the iron becoming 
harder by the greater pressure required in forming the screw thread when the dies 
are old and blunt than when new and sharp. 

52. The strength of screw-bolts is found to be in proportion to their relative 
areas, there being only a slight difference in favor of the smaller compared with the 
larger sizes, instead of the very material difference previously imagined. 

55. Screwed bolts are not necessarily injured, although strained nearly to their 
breaking point. 

54. A great variation exists in the strength of iron bars which have been cut and 
welded ; whilst some bear almost as much as the uncut bar, the strength of others is 
reduced fully a third. 

55. The welding of steel bars, owing to their being so easily burned by slightly 
overheating, is a difficult and uncertain operation. 

56. Iron is injured by being brought to a white or welding heat, if not at the 
same time hammered or rolled. 

57. The breaking strain is considerably less when the strain is applied suddenly 
instead of gradually, though some have imagined that the reverse is the case. 

58. The contraction of area is also less when the strain is suddenly applied. 

59. The breaking strain is reduced when the iron is frozen ; with the strain 
gradually applied, the difference between a frozen and unfrozen bolt is lessened, as 
the iron is warmed by the drawing out of the specimen. 

60. The amount of heat developed is considerable when the specimen is sud- 
denly stretched, as shown in the formation of vapor from the melting of the layer 
of ice on one of the specimens, and also by the surface of others assuming tints of 
various shades of blue and orange, not only in steel, but also, although in a less 
marked degree, in iron. 

61. The specific gravity is found generally to indicate pretty correctly the 
quality of specimens. 

62. The density of iron is decreased by the process of wire-drawing, and by the 
similar process of cold rolling, instead of increased^ as previously imagined. 

63. The density in some descriptions of iron is also decreased by additional hot- 
rolling in the ordinary way ; in others the density is very slightly increased. 

64. The density of iron is decreased by being drawn out under a tensile strain, 
instead of increased, as believed by some. 



2/6 



CAST IRON IN TENSION. 



[Art. 33. 



65. The most highly converted steel does not, as some may suppose, possess the 
greatest density. 

66, In cast steel the density is much greater than in puddled steel, which is 
even less than in some of the superior descriptions of wrought iron. 



Art. 33. — Cast Iron, 

Coefficient of Elasticity and Elastic Limit. 

Cast iron is a material of much less value to the engineer 
than wrought iron, and consequently has been the subject of 
much less experimental investigation. 

The following table (Table I.) contains values of the co- 
efficient of tensile elasticity for three (Nos. i, 2 and 3) differ- 
ent irons used in the fabrication of cast-iron cannon. They 
are computed by the aid of Eq. (i), Art. 2, from data con- 
tained in " Reports of Experiments on the Properties of 
Metals for Cannon," etc., by the late Captain T. J. Rodman, 

TABLE I. 





NO. I. 


NO. 2. 


NO. 3. 


. NO. 4. 


w. 


E. 


E. 


E. 


E. 


1,000 
2,000 

3,000 

4,000 

5,000 
10,000 
15,000 
20,000 
24,000 


28,011,000 
28,011,000 
25,000,000 
22,962,000 
23,031,000 
20,960,000 
16,773,000 
13,384,000 
10,150,000 


50,000,000 
28,571,000 
23,810,000 
22,727,000 
20,833,000 
17,000,000 
13,204,000 
7,370,000 
3,454,000 


33,333.000 
28,571,000 
27,273,000 
25,000,000 
23,810,000 
20,000,000 
17,241.000 
14,085,000 
11,060,000 


25,000,000 
16,667,000 
15,000,000 
15.385,000 
13,889,000 
12,195,000 
10,000,000 
8,000,000 





W and E are expressed in pounds per square inch. 

U. S. A. The iron was an excellent charcoal gun iron, and the 
specimens were from 30 to 35 inches long turned to a diameter 
of 1.382 inches. The data were selected at random (pages 158, 



Art. 33.] 



ELASTICITY. 



277 



212 and 228 of the work cited) from the large amount accumu- 
lated by Captain Rodman. 

Column No. 4 contains values of E given by Wm. Kent, 
M. E. (Van Nostrand's Magazine, Vol. 20) ; they belong to a 
piece of cast iron i J inches in diameter and 5 inches long. 

The left-hand column, headed "■ W," gives the stress per 
square inch, while the three columns **ii" give the correspond- 
ing ratios between stress and strain for the three different 
irons. Such ratios are the *' coefficients of elasticity," properly 
speaking, below the elastic limit only. It will be observed, 
however, that none of these specimens can really be con- 
sidered to possess an elastic limit, unless possibly No. i, 
whose elastic limit may be taken at, or a very little above 
2,000 pounds per square inch. 

In No. I first permanent set was observed at 4,ocx) pounds per square inch. 
In No. 2 first permanent set was observed at 4,000 pounds per square inch. 
In No. 3 first permanent set was observed at 8,000 pounds per square inch. 



































































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F-ig.l 
Fig. I represents graphically the results of the experiments 



2/8 



CAST IRON IN TENSION, 



[Art. 33. 



on specimen No. 2. The constantly varying value of the ratio 
between stress and strain is shown in a very evident manner 
by the continually varying inclination of the curve. The 
strains (stretches) are laid down as if belonging to a bar 1,000 
inches long. 

The following results are deduced by B. B. Stoney (Theory 
of Strains in Girders and similar Structures, p. 369) from ex- 
periments by Eaton Hodgkinson on a bar of English cast iron 
10 feet long. 

W = 2,240 pounds per square inch E ■= 13,603,520 pounds per square inch. 



W = 4,480 

W= 6,720 

W= 8,960 

W = 11, 200 

W— 13,440 

W= 14,560 



.E = 13,260,800 
.E = 12,382,720 
.E = 11,596,480 
.E = 10,843,840 
.E = 9,856,000 
.E = 9,549,120 



These results show a limit of elasticity at about 6,000 
pounds per square inch ; they also show much smaller values 
of £ than those given in Table I. This last disagreement is 
undoubtedly due, to a great extent, to the fact that the values 
of E in Table I. probably a// belong to fine charcoal iron fabri- 
cated for a special purpose, while the others do not. 

If A = extension, or stretch in inches of a cast-iron bar 
when acted upon by a force W (in pounds), and if / represents 
the length of the bar in inches, Mr. Hodgkinson deduced the 
following formulae from his experiments : 



A = / {.00239628 — '\/.ooooo5742i5 — .000000000343946?^} . (i) 
For bars 10 feet long: 



Permanent set, in inches = .0193A + .64/V (2) 



Although the preceding results are only a few of a great 



Art. 33.] 



ULTIMATE RESISTANCE. 



279 



many similar results that may be computed in the same man- 
ner, yet they give a fair representation of the general character 
of the elastic properties of cast iron. The metal is seen to be 
very irregular and unreliable in its elastic behavior. A large 
portion of the material can scarcely be said to have an elastic 
limit, although no apparent permanent set takes place under a 
considerable intensity of stress ; in other words, although per- 
haps all tested specimens resume their original shape and 
dimensions for sm.all intensities of stress, yet the ratio between 
stress and strain is seldom constant for essentially any range 
of stress. 

Ultimate Resistance. 

On page 5 of Captain Rodman's "■ Reports " are given the 
following densities and ultimate tensile resistances, expressed 
in pounds per square inch, of 16 specimens of warm-blast, 
charcoal Greenwood and Salisbury iron, taken from preliminary 
castings of second and third fusion pigs : 



ISITY. ULT. RESIST. 

184 33,079 

I9S 31,384 

307 35,486 

099 23,776 

304 31,317 

273 42,884 

272 38,993 

219 25,372 



DENSITY. ULT. RESIST. 

7-210 22,547 

7.172 28,518 

7-159 36,373 

7-137 33,268 

7. 106 22,290 

7. TOO 22,779 

7.109 22,888 

7-191 23,873 



Again, Table II. is taken from page 261 of the same " Re- 
ports." The results are for specimens from trial castings of 
second-fusion pigs. The ultimate resistance is in pounds per 
square inch, while the strains are for an inch of length. 

" Uli. Ext.'' is the ultimate extension, or stretch, just be- 
fore fracture, for one lineal inch. The specimens were 30 
inches long and 1.382 inches in diameter. 



280 



CAST IRON IN TENSION. 



[Art. 33. 



TABLE II. 



SPECIMEN. 


DENSITY. 


ULT. EXT. 


ULT. RESIST. 


Ao 


7.267 


. 00303 


30,117 


Ai 


7.274 


.00334 


31,681 


Bo 


7.178 


.00291 


23,617 


Bi 


7.202 


.00161 


24,260 


Co 


7-255 


.00287 


28,220 


Cr 


7.280 


.00382 


27,147 


Do 


7.221 


.00424 


25,627 


Di 


7.230 


.00223 


24,767 



On page 42 of ** Reports of Experiments on Metals for 
Cannon," Major Wade gives the following for 15 proof bars 
cast with 8- and lO-inch guns and 6-pounder trial guns, at South 
Boston, 1844 : 

Greatest resistance per square inch 31,027 pounds. 

Mean " " " " 27,232 " 

Least '* " " " 22,402 " 



He states that these specimens show the general quality of 
the iron used at that time. 

Again, on page 179 of the same " Reports,*' Major Wade 
gives for 25 specimens from 32-pounder cannon made at West 
Point foundry in 1850 : 

Greatest resistance per square inch 36,728 pounds. 

Mean " " " " 32,023 " 

Least " " " " ■ 28,990 *' 

He states that the character of this iron was *' that of good 
foundry iron, of the different grades of Numbers I, 2, and 3 ; " 
it was composed of first, or first and second fusion pigs. 

The preceding results give correct representations of the 
character of the best quality of American cast iron, produced 
for use in cases requiring such a metal. 



Art. 33-] ULTIMATE RESISTANCE. 28 1 

Three specimens, turned down to a diameter of about 
0.625 inch, taken from the iron used in the Boston water 
mains, and broken at the Warren Foundry, Phillipsburg, N. J., 
gave the following ultimate resistances in pounds per square 
inch : 

13.070 15.470 18,300 

As with all material, the character of cast iron affects, to a 
great extent, its resistance ; i. r., whether it is fine or coarse 
grained, gray or white, etc. It (the resistance) also depends 
upon the character of the ore from which it is produced. 

Major Wade (" Reports," pages 378 and 388) shows that 
the cold-blast iron which he tested gave much higher resist- 
ance than the hot-blast metal. 

It is to be remembered that all the specimens from which 
the preceding results were deduced were what may be called 
" small specimens." Specimens with several square inches in 
area of normal section would probably give somewhat different 
results. 

It is interesting to observe that, in experimenting upon 
cast-iron cannon. Major Wade (" Reports," pages JJ and 78) 
found that, water was forced through the "pores" of the 
metal of one cannon at a pressure of 7,000 pounds per square 
inch, and through those of another with thicker metal (thick- 
ness equal to radius of bore) at a pressure of 9,000 pounds per 
square inch. 

Capt. Rodman ('* Reports," page 262) forced water through 
the pores of the metal of cylinders 5 inches long, i inch thick, 
and I inch bore, at pressures ranging from 15,276 to 25,464 
pounds per square inch. 

The experiments of Eaton Hodgkinson ('* Experimental 
Researches on the Strength and Other Properties of Cast 
Iron "), on English metal gave the following resistances in 
pounds per square inch : 



282 CAST IRON IN TENSION. [Art. 33. 

Cannon iron No. 2, hot blast I3»505 pounds. 

Cannon iron No. 2, cold blast 16,683 " 

Cannon iron No. 3, hot blast 17,755 " 

Cannon iron No, 3, cold blast 14,200 " 

Devon (Scotland) iron No. 3, hot blast 21,907 " 

Buffery iron No. i, hot blast 13,434 " 

Buffery iron No. i, cold blast 17,466 '* 

Coed-Talon iron No. 2, hot blast 16,676 " 

Coed-Talon iron No. 2, cold blast 18,855 '* 

Low Moor iron No. 3 14,535 " 

Mixture 16,542 " 

Several of these results are the means of those of a number 
of tests. The areas of the normal sections of the test speci- 
mens varied from 1.54 inches to 4.27 inches, being considerably 
larger than those of the specimens tested by Major Wade and 
Captain Rodman. 

The characteristic fracture of cast iron is granular and crys- 
talline, with very little (scarcely perceptible by the unaided 
eye) reduction of area or elongation. Fracture takes place 
suddenly and without warning, and its ultimate resistance is 
influenced by many causes whose action may not be observed 
by any ordinary means ; for these reasons, it is a treacherous 
and unreliable material in tension, as indeed any brittle ma- 
terial must be. 



Effect of Reinclting, 

Crude pigs are said to be '''first-fusion " metal. 
Once remelted pigs produce ''' second-fusion'' iron. 
Twice remelted pigs produce '' tJiird-fusion " iron. 

etc., etc., etc. 

On page 237 of Major Wade's "• Reports," the following 
values are given for Greenwood first-fusio7i iron (iron in orig- 
inal pigs) : 



Art. 33.] EFFECT OF REMELTING. 283 

ULT. RESIST. IN POUNDS 
PER SQ. IN. 

No. I iron 15,129 (mean of 3 tests). 

No. 2 iron 27,153 (mean of 2 tests). 

No. 3 iron 34i923 (mean of 4 tests). 

** No. I is the softest gray iron, 

" No. 2 is intermediate, 

" No. 3 is the hardest gray iron." 

Again on page 240 : 

ULT, RESIST, 

Greenwood, No. i, ist fusion 20,900 pounds per sq. in. 

Greenwood, No. i, 2d fusion 30,229 pounds per sq. in. 

Greenwood, No. i, 3d fusion 35,736 pounds per sq. in. 

Guns cast from 3d fusion 33, 81 5 pounds per sq, in. 

The last result is a mean of four tests. 
Finally on page 242 : 

Nos. I and 2 mixed | '^ ^"'^°" ^7,588 pounds per sq. in. 

( 3d fusion 40,987 pounds per sq. in. 

Nos. I, 2, and 3 mixed. \ "^ ^^^^^^ 37,789 pounds per sq. in. 

' 3d fusion 32,485 pounds per sq. in. 

It is seen that " the softest kinds of iron will endure a 
greater number of meltings with advantage, than the higher 
grades." The greatest ultimate resistance, in pounds per 
square inch, is obtained with : 

No, I iron at the 4th fusion, 

Nos. I and 2 mixed at the 3d fusion, 
Nos, I, 2 and 3 mixed at the 2d fusion. 

These results probably indicate about the limits to which 
the remelting of this iron could be advantageously carried. 

On page 279 of the same '' Reports," is given the result of 
the test of a specimen of third-fusion iron, of a mixture of No.s. 
I, 2 and 3, taken from a gun. The ultimate resistance found 



284 



CAST IRON IN TENSION. 



[Art. 33. 



was 45,970 pounds per square inch ; a most remarkable speci- 
men of cast iron. 



Effect of Continued Fusion. 

Major Wade (" Reports," pp. 38-41) tested the effects of 
continued fusion on different grades of iron, both in relation to 
transverse and tensile resistance. 

The general result was an increase of tensile resistance up 
to 3^ hours in fusion, which was the longest period tried. 

The following results are taken from pp. 40 and 41 of the 
" Reports." 



'^ 



TIME IN FUSION. 



ULT. RESIST. 



Stockbridge. 



Proof bars. . - 



No. 5. 
No. 6. 

No. 7. 
No. 8, 



lo-inch Howitzer, 2d 
fusion from pigs. 




hour 17,843 pounds per square inch. 

" 20,127 " 

" 24.387 " 

" 34,496 " 

" 25,969 " 

*' 29,143 " 

" 27,755 " 

" 30,039 

" 15.861 " 

*• 20,420 " " 

" 24,383 " 

" 25,773 " 



These tests show well the effect of continued fusion for a 
period not exceeding 3.75 hours. 

Effect of Repetition of Stress. 

Capt. Rodman (" Reports," p. 262) experimented on the 
effect of repeated stresses with the following results : 



SPECIMEN. 



Ac broke at 2301st repetition of 22,000 pounds per square inch. 



Ac 
Bo 
Bo 
Co 
Co 
Do 



282d 
252d 

150th 
651st 

457th 

I72d 



" 26,000 
20,000 
20,000 
22,500 
23,500 
21,600 



Art. 33.] REPETITION OF STRESS. 285 

The repetition of the letters representing the specimen 
indicates that duplicates were tested. 

A reference to Table II. will show what single loads per 
square inch broke the same irons, and a comparison of the two 
will exhibit the " fatigue " of the metal. 

On pages 166 and 167 he also gives some very interesting 
results of intermittent repetitions of stresses. He subjected a 
cylinder of cast i-ron, 1.382 inches in diameter and 35 inches 
long to intermittent repetitions of 15,000 pounds per square 
inch (about three-quarters of its ultimate resistance) as follows: 
250 repetitions, then a rest of 40 hours ; next, 375 additional 
repetitions, then a rest of 30 days; next, 155 additional repe- 
titions, then a rest of 29 days ; next, 1,020 additional repe- 
titions, then a rest of 26 days; finally, 156 additional repe- 
titions followed by breakage at the 1,956th repetition. In 
every case ''■ rest " signifies entire freedom from load. Capt. 
Rodman's table gives a detailed account of these experiments. 
He remarks upon them as follows : " The most interesting 
point ... is the fact that at every interval of rest, of any 
considerable time, the permanent set, and the extension due to 
the last previous application of the force, diminished. And in 
some instances it required some fifty repetitions to bring up 
the extension and set to the same points where they had been 
at the beginning of the period of rest ; thus indicating clearly 
that the specimen was partially restored, by the interval of 
rest, from the injury which it had received ; and that it endured 
a greater number of repetitions, owing to the intervals of rest, 
than it would have done had the repetitions succeeded each 
other continuously, and at short intervals of time." 

These experiments show the *' fatigue " of cast iron and 
the increase of the ratio of stress over strain produced by 
** rest " — so far as tensile stress is concerned. 

An examination of the tables also shows that in any series 
of repetitions, between any two consecutive rests, both the 
extension and set were constantly incrcasi?ig, consequently, 



286 



STEEL IN TENSION. 



[Art. 34. 



that the ratio of stress over strain was constantly decreas- 
ing. 

Effect of High Temperatures, 

A few experimental results bearing on this point will be 
found in Table IX. of Article 35. 



Art. 34.— Steel. 
Coefficient of Elasticity. 

The great number of the varieties and grades of " steel " 
renders possible the existence of a correspondingly great num- 
ber of the mechanical quantities and coefficients used in its 
consideration in connection with the " Resistance of Mate- 
rials." In every case, therefore, the kind and character of the 
steel on which experiments are made, should be stated. In 
some cases, however, this is impossible. 

TABLE A. 



Coleman, Rahm & Co., Pitts 

burg 

Am. Tool Steel Co., Brooklyn 
Butcher & Co., Philadelphia 



COEFFICIENT OF ELASTICITY. 



Greatest. Mean 



31,000,000 

20,325,000 
24,200,000 
22,600,000 
32,150,000 
25,100,000 
25,700,000 



29,500,000 
25,600,000 
15,683,400 
17,154,000 
17,298,000 
22,835,000 
21,145,000 
22,712,000 



Least. 



About 27,000,000 



28,000,000 

13,324,600 

11,950,000 \ 

11,115,000 

17,605,000 

17,350,000 

21,400,000 



" Very poor steel." 

From chrome steel 

staves. 
Chrome steel from ingot. 
Carbon rivet steel. 
Carbon steel. 
Carbon steel. 
Carbon steel staves. 



Table A contains coefficients of tensile elasticity for the 
different grades of steel shown. These results were obtained 
from tests made in connection with the construction of the 



Art. 34.] 



COEFFICIENT OF ELASTICITY. 



287 



St. Louis steel arch, and have been taken from Prof. Wood- 
ward's history of that structure. 

These coefficients were determined for cylindrical speci- 
mens varying from 0.5 inch to i.oo inch in diameter and 3.00 
to 6.00 inches in length. 

In Table I. are contained the coefficients of elasticity of 
the hardened and tempered steel wire (see Table XXII.), sup- 
plied by the different makers named, in response to the call 
for bids for the steel cable wire for the New York and Brook- 
lyn suspension bridge. (Washington A. Roebling's " Report," 
1st Jan., 1877). 

In the same " Report," page 72, the specifications state : 
" The elastic limit must be no less than -^^-^ of the breaking 
strength. . . . Within this limit of elasticity, it must 
stretch at a uniform rate corresponding to a modulus of elas- 
ticity of not less than 27,000,000 nor exceeding 29,000,000." 

TABLE I. 



PRODUCER. 



J. Lloyd Haigh 

Cleveland Rolling Mills. ...... 

Washburne & Moen 

Sulzbacher, Hymen, Wolff & Co 

Jno. A. Roebling's Sons Co 

Carey & Moen 



COEFFICIENTS OF ELASTICITY. 



Greatest E. 



29,817,067 
30,142,026 
29,757,300 
30,389,946 
30,231,929 
31,261,041 



Least E. 



28,815,797 
28,917,715 
28,887,006 
29,103,238 
28,788,619 
29,418 025 



NUMBER OF 
TESTS. 



12 

6 
6 
6 

13 
12 



Table I. gives the greatest and least results of these 
tests in pounds per square inch, in the columns headed *' ^/* 



288 



STEEL IN TENSION. 



[Art. 34. 



together with the number of tests of the product of each 
maker. All the wire was No. 8, Birmingham gauge ; /. e., 
0.165 inch in diameter. 

It not evident from the " Report " whether these values 
were obtained for some particular intensity of stress, or 
whether they are mean values for the entire range below the 
elastic limit. 

TABLE II. 



KIND OF STEEL. 



Hammered Bessemer from 

Hogbo (round) 

Hammered Bessemer from 

Hogbo (square) 

Hammered Bessemer from 

Hogbo (square) 

Rolled cast steel from Wik- 

manshyttam (round) . . . 
Hammered cast steel from 

F. Krupp (round) 

Rolled puddled steel from 

Surahammar (square). . . 
Rolled puddled steel from 

Surahammar (square). . . 



SPEC. 








CARBON. 


SET. 


GRAY. 






7.832 


1-35 




7.850 


1.26 


0.004. 


7.849 


1.05 


0.014 


7.832 


1.22 


0.021 


7.843 


0.61 


0.0008 


7.781 


0.66 




7.828 


0.56 


0.027 



30,124,180 
30,604,520 

31,222,100 

31,359.340 



29,918,320 



0.003 
0.006 
0.000 



0.004 
0.0,15 



31,839,680 
30,535,900 
31,496,580 



32,114,160 
30,330,040 



Table II. contains coefficients of tensile elasticity for the 
steels named, as determined by Knut Styffe (*' Iron and Steel," 
pages 146 and 147). He pursued the following method : Let 
/' and /represent the stretch, or strain, for unit of length of a 
bar for the two intensities /' and /. By the principles estab- 
lished in Article 2 : 



/ = ^ , and 
E 



E E 



Hence, 



Art. 34.] COEFFICIENT OF ELASTICITY. 289 

^=^ 0) 

Eq. (i) really gives a kind of *' mean " coefficient, for it is 
based on the assumption that E is the same for different in- 
tensities of stress. It will be seen in Table III. that this is 
sometimes far from true. 

The columns ** carbon " and " set '* contain per cents of 
those quantities. E is the coefficient of elasticity before the 
bar is heated, and E' the same quantity after the bar had been 
heated to " slight redness." ** Set " is the permanent elonga- 
tion (in per cents of original lengths) just before E or E' was 
measured. The test specimens were small bars varying in 
original area from 0.1015 square inch to 0.2065 square inch. 

The experiments of Styffe showed that *' by such mechani- 
cal operations as stretching, hammering, etc.," the coefficient 
of elasticity may be diminished ; *' whilst by a moderate heat, 
or still better by a glowing heat, it may be increased." 

Table III. has been computed from data obtained by 
David Kirkaldy during his experiments on Fagersta steel 
plates ('' Experimental Inquiry into the Properties of Fagersta 
Steel," series D, Part i). The test specimens were, in the 
clear, 2J^ inches wide and 100 inches long. The thickness is 
given in the horizontal row, as shown. The values of the co- 
efficient of elasticity [E) are the greatest and least, in pounds 
per square inch, for the various intensities *'/," for five unan- 
nealed j^j %, ^, J^ and 5^-inch (nominally) plates and five 
similar annealed ones. 

These show very irregular elastic behavior. The 5^ inch 
annealed specimen is the only one which can properly be con- 
sidered as possessing a true *' coefficient of elasticity " (about 
29,000,000 pounds per square inch) above the stress intensity 
of 10,000 pounds, the ratios of stress to strain are so very 
variable. Prof. Bauschinger's "stretch-limit" is clearly shown, 

for the different specimens, at that point of stress where the 
19 



290 



STEEL IN TENSION. 



[Art. 34. 



TABLE III. 



>. 


UNANNEALED. 


ANNEALED. 


Greatest E. 


Least E. 


Greatest E. 


Least E. 


10,000 
14,000 
18,000 
22,000 
26,000 
30,000 

34,000 
38,000 
42,000 
46,000 
50,000 


45,455,000 
38,889,000 
36,000,000 
34,375,000 
33,333,000 
31,915,000 
30,631,000 
29,008,000 
13,084,000 
4,670,000 
2,294,000 


33,333,000 

30,435,000 

29,032,000 

28,205,000 

25,000,000 

1,714,000 

1,107,500 

821,000 


37,037,000 
34,146,000 
32,727,000 
31,429,000 
30,952,000 
29,412,000 
4,151,000 
1,214,000 


29,412,000 

29,167,000 

29,032,000 

28,947,000 

20,968,000 

1,765,000 

1,066,000 

805,000 




















Thickness. 


0. 125 inch. 


0.380 inch. 


0.255 inch. 


0.625 inch. 



values of E are almost annihilated. In these four specimens 
the first permanent sets were noted at 40,000, 20,000, 30,000 
and 20,000 pounds per square Inch respectively. 

In 1868 and 1870 the "Steel Committee" of the British 
Institution of Civil Engineers made some valuable experiments 
on different grades of steel. The following values of E^ in 
pounds per square inch, are computed from data established 
by that committee : 

Eighteen specimens, 50 inches long and 1.382 incites in diame- 
ter, of Bessemer steel tires, axles and rails (14 hammered and 4 
rolled), gave : 

Greatest E r= 19,310,000 pounds per square inch \ 

Mean E := 18,211,000 pounds per square inch > . . (2) 

Least E = 17,231,000 pounds per square inch ) 

Ten hammered samples of crucible steel tires, axles and rails, 
and one of rolled axle, all with preceding dimensions, gave : 



Art. 34.] COEFFICIENT OF ELASTICITY. 29 1 

Greatest E =■ 19,310,000 pounds per square inch \ 

Mean -£"= 17,778,000 pounds per square inch >• . . (3) 

Least ^ = 16,232,000 pounds per square inch) 

All these are evidently for very soft steels, none of whose 
tensile resistances exceeded 91,700 pounds per square inch. 

The following results are computed from samples ten feet 
long and one and one half inches in diameter: 

Eleven samples of Bessemer steel gave : 

Greatest E =. 29,867,000 pounds per square inch \ 

Mean ^ = 28,718,000 pounds per square inch >- . . (4) 

Least E -^ 27,317,000 pounds per square inch ) 

Ni?ieteen specimens of crucible steel {chisel-rods^ gim-barrels, 
etc.), gave : 

Greatest E ^= 29,867,000 pounds per square inch j 

Mean ^ = 28,718,000 pounds per square inch >• . . (5) 

Least £ = 26,654,000 pounds per square inch ) 

These last were much harder steels, with ultimate resist- 
ance varying from 75,300 to 118,300 pounds per square inch. 

All the results in Eqs. (2), (3), (4) and (5) are calculated for 
the strains at the so-called " elastic limit." It is probable that 
considerably larger values would be obtained for the ratio {E) 
between stress and strain at m.uch lower intensities of stress. 

Prof. Alex. B. W. Kennedy (London " Engineering," Vol. 
XXXI., 1881) determined the coefficients of tensile elasticity 
of specimens of mild steel plates containing about 0.18 per 
cent, of carbon, and of some specimens of still milder rivet 
steel. 

Twelve specimens of plates (i^ x ^; 4X ^;2x ^; 
3^ X ^ ; and 2y^ X y2, all in incites) gave : 

GREATEST. MEAN. LEAST. 

33,670,000 29,882,000 25,440,000 

all m pounds per square inch. 



292 STEEL IN TENSION. [Art. 34. 

EigJit other specimens of the same plates gave : 

GREATEST. MEAN. LEAST, 

31,940,000 29,001,000 26,110,000 

all in pounds per square inch. 

As a rule, the thinner plates gave the higher values of E. 
There were, however, some marked exceptions. 

Eleven specimens of \\ inch round rivet steely turjted to about 
I inch diameter; two each of \\ arid ijV ^'^^^>^^ rounds turned to \ 
and \ inch diameter^ respectively^ gave : 

GREATEST. MEAN. LEAST. 

31,750,000 30,670,000 29,790,000 

all in pounds per square inch. 



Hay Steel. 

Some experiments on three different bars of the Hay steel 
used in the bridge at Glasgow, Missouri, by Gen. Wm. Sooy 
Smith, gave the following results ('' Annales des Fonts et 
Chaussees," Feb., 1881) : 

Experiment No. i. 

A bar of rectangular section 2.09 x i.i inches, reduced by 
hammering from a bar 2.6 inches square, was subjected to dif- 
ferent intensities of stress varying from about 20,500 to 54,000 
pounds per square inch, at which the following values of the 
coefficients of elasticity (in pounds per square inch) were 
found : 

GREATEST. MEAN. LEAST. 

32,900,000 28,824,000 26,094,000 

At 54,000 pounds per square inch there was a *' trace " only 



Art. 34.] COEFFICIENT OF ELASTICITY. 293 

of permanent elongation or set. The length of this bar, be- 
tween the observation marks, was about 38.5 inches. 

Experiment No. 2. 

A round bar 1.04 inches in diameter was subjected to a 
stress of about 51,200 pounds per square inch, with a stretch of 
1.66 millimetres per metre, at which a *' trace " only of perma- 
nent set was observed. The resulting coefficient of elasticity 
was : 

E = 30,857,000 pounds per square inch. 

The distance between observation marks was about 18.7 
inches. 

Experime7it No. 3. 

A bar 5.2 x 1.34 inches was subjected to a stress of about 
49,200 pounds per square inch, with a trace only of permanent 
set and a strain of 0.00171 metre per metre. Consequently 
the resulting coefficient of elasticity was : 

E = 28,764,000 pounds per square inch. 

These experiments show that the coefficient of elasticity of 
Hay steel is not essentially different from that of other mate- 
rial of the same class. 

In his " Report on the Renewal of Niagara Suspension 
Bridge," Mr. Leffert L. Buck, C. E., gives the following values 
for the Hay steel used in that work : 

GREATEST. MEAN. LEAST. 

30,830,000 28,000,000 26,400,000 

all in pounds per square inch. These results are for eighteen 
experiments on small specimens. 



294 



STEEL IN TENSION. 



[Art. 34. 



Ultimate Resistance and Elastic Limit. 

In this section, it is to be observed, the " elastic limit " is 
seldom that point at which the coefficient of elasticity (stress 
over strain) ceases to be essentially constant, but more nearly 
Prof. Bauschinger's " stretch-limit," at which the increment of 
strain, due to a constant increment of "Stress, very suddenly 
increases, involving a correspondingly great permanent set. 

TABLE III^. 







ULTIMATE 


. RESISTANCE, POUNDS 









PER SQUARE INCH. 




MAKERS. 


Z 

< 

u 








REMARKS. 




Greatest. 


Mean. 


Least. 




Coleman, Rahm & Co., Pitts- 












burg . 


3 


118,400 


91,200 
106,500 


74,000 


" Very poor steel." 
For lathe tools. 


Am. Tool Steel Co., Brooklyn. 


I 


Butcher & Co., Philadelphia.. 


7 


144,300 


112,100 


93,500 




Park Bros. , Pittsburg 


2 


118,000 


118,000 


118,000 




Steel Works, New York 


I 




85,400 






Jessup, Sheffield, Eng 


4 


86,000 


78,500 


74,000 


J 2 Annealed. 
1 2 Unannealed. 


Anderson & Woods, Pittsburg. 


2 


100,000 


100,000 


100,000 




Coleman, Rahm & Co., Pitts- 












burg. 


I 




68,000 






Miller, Barr & Parkin, Pitts- 




burg 


I 




90,000 




Annealed. 


Miller, Barr & Parkin, Pitts- 












burg 


2 


103,200 


102,200 


101,200 


Unannealed. 


Hussay, Well- «& Co., Pitts- 












burg 


3 
7 


128,000 


126,500 
75,450 


125,000 
45,000 


Steel plate. 

Cast "machinery steel." 


Brown & Co., Pittsburg 


81,300 


Thos. Firth, Sheffield, Eng... 


3 


113,900 


112,600 


112,000 


" Gun metal." 


Butcher & Co., Philadelphia.. 


4 


110,100 


103,500 


99,200 


Chrome steel stave. 


ik 


" 


4 


107,500 


106,000 


103,500 


Chrome steel slave. 


Ik 


t Ik 


3 


151,000 


148,700 


147,000 


Chrome steel ingot. 


li 


k kk 


5 


129,000 


99,900 


69,800 


Carbon rivet steel. 


Ki 


I kk 


6 


128,300 


98,300 


65,300 


Carbon steel. 


kk 


' *' 


5 


142.000 


120,100 


100,000 


Carbon rivet steel. 


kk 


k kk 


5 


143,600 


139,300 


136,300 


Chrome steel stave. 


kk kk kk 


4 


135,400 


119,200 


111,700 


Chrome steel stave. 


N. Y. Chrome Steel Co 


16 


193,260 


146,400 


115,800 


Chrome steel bar. 


Parks Bros., Pittsburgh ^z 


3 


131,864 


119,500 


109,500 


Steel normal untemp. 


kk kk kk 1 CJ • 

kk » u Jf« 


3 


227,500 


217,400 


201,341 


Temp, in oil at 82" F. 


3 


176,100 


165,500 


152,500 


Temp, in water at 79" F. 


3 


150,500 


141,900 


132,700 


Temp, in water at 79" F. 



Art. 34.] 



RAIL STEEL. 



295 



Table \Wa is condensed from Prof. Woodbury's history of 
the St. Louis arch. The last four results are from the experi- 
ments of Chief Engineer Shock, U.S.N., while the " N. Y. 
Chrome Steel Co." result is from Kirkaldy's tests. 

The diameters of the (circular) specimens varied from 0.357 
inch to 1. 00 inch, and their lengths from 3 to 12 inches. The 
elastic limit varied from 45 to 55 (nearly) per cent, of the ulti- 
mate resistance. 

TABLE IV. 
Rail Steel. 





1 


JN ANNEALED. 






ANNEALED. 




CARBON, 


NO. 


















T. 


E. L. 


STR. 


T. 


E.L. 


STR. 


PERCENTS. 


8 


79,625 


37,625 


ig.6 


78,250 


35,750 


20.5 


0.324 


8 


81,250 


36,625 


15.6 


77,375 


37,125 


14.7 


0.379 


4 


72,750 


32,250 


22.5 


70,750 


29,250 


22.0 


0.308 


4 


76,750 


38,250 


II-5 


74,000 


30,500 


9-5 


0.438 


4 


75,750 


36,500 


12.8 


73,750 


34,000 


15.2 


0.405 


4 


83,500 


37,000 


14-5 


80,750 


40,000 


15.2 


0.384 


8 


1^,31S 


36,625 


17-5 


70,500 


33,875 


19.2 


0.282 


8 


19^^15 


38,000 


14-5 


78,000 


36,875 


14.4 


0.381 


4 


70, 500 


34,250 


17.0 


69,500 


30,500 


17-5 


0.367 


4 


82,000 


40,250 


12.7 


76,000 


38,500 


13.2 


0.394 


4 


78,000 


36,750 


10.7 


74,250 


33,250 


13-7 


0.378 


4 


77,000 


40,2';o 


I5-0 


76,000 


35,750 


14.5 


0.388 


24 


74,542 


35,833 


1S.9 


72,958 


33,167 


19.8 


0.314 


24 


80,167 


37,953 


14. 1 


76,792 


36,167 


13-5 


0.392 


8 


76.875 


36,625 


II. 7 


74,000 


33,625 


14.4 


0.391 


8 


80,250 


38,625 


14.7 


78,375 


37,875 


14.8 


O.3S6 


32 


75,125 


36,031 


17. 1 


73,219 


33,281 


18.5 


0.334 


32 


8o,i83 


38,125 


14.2 


77,188 


36,594 


13-8 


0.390 



Table IV. contains the results of one hundred and ninety- 
two tests of specimens from steel rails which had been in use 
on the Penn. R. R. during periods of time of different lengths, 
These results were given by Charles B. Dudley, Ph.D., Chemist 
to Penn. R. R. Co., in his " Report to the Supt. of Motive 



296 



STEEL IN TENSION. 



[Art. 34. 



Power," and published in the Journal of the Franklin Institute, 
March, 1 881. 

The specimens were circular and turned to a diameter of 
0.75 inch between shoulders five inches apart. 

The following is the notation : 

"iV<?." is the number of specimens for which the other 

quantities are the average. 
" Z." is the ultimate tensile resistance in pounds per 

square inch, 
"ii. Lr is the elastic limit in pounds per square inch. 
" Str^ is the per cent, of original length of ultimate 

elongation or stretch {i. e.y at instant of rupture). 

TABLE V. 
Bars Unannealed. 



CARBON. 


STRESS IN POUNDS PER SQUARE INCH. 


PER CENTS. 


Per cent. 


Ultimate. 


Elastic Limit. 


Str. 


Cont. 


0.30 
0.30 
0.30 
0.30 
0.30 


94,760 1 1 
95,380 1 xF, 
95,830 \ °^ 
96,020 ll 
94,970 J ^• 


55,712 ~ 
56,009 
55.120 
55,830 
55.512 J 


in 

CO 

vO 
in 

> 

< 


^5.0 ": 
12.9 1 J 

15.3 y II 

14-5 • 

13.8 J ^ 


30. ^ 
26. 
31. 
27. 
29. J 




CO 

^ II 

> 

< 


0.50 
0.50 
0.50 
0.50 
0.50 



112,340 "1 
112,470 ^'■ 

111,980 y ^ 
113,320 II 
113,040 J ^ 


65-790 ] 8n 

66,040 xn 

66,160 y ^ 
65,550 1 I' 
65,980 j ^ 


10.8 ^ ^ 
8.9 ! 2 

10.5 1^ ii 

10.9 1 ^ 
9-4 J < 


19. ' 

16. 

22. 

21. 

20. 




ON 
M 

- 1! 

> 

< 



"■ Sir."" is the elongation of original length. 
" Cont,'' is the contraction of original section. 



Art. 34.] 



E YE-BARS. 



297 



0. 


u 


Id 


% 


X 








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a 


U. 


h 


a 


U 


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X 



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0) 


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^ 


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rC 











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rp-c 


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cn 


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M rj- M 



Ph 






o •" •« 



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S-K = -Ay 
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< Z 2 

w 



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1 

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W 

CL. 

K 
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fa 

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g i s 

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fa -1 

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s-6 = -Ay 

M en CO 

CO a^ w 

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£-z = 


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1-5 


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N 

r->. i--» CO 
sO ir> r-- 

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= -Ay 





^ 

en 




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■isL'zS = -Ay 


•l7i9'9S = 

J- 


r -Ay 




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q_ M_ M^ 

-r rf rf 
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M* T^F C!^ 
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en 

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en 

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•?3sdn 


•pailoH 


•pappAV 




fa 


CO 

2 


In 
Z 

S 
5 


•910H UIJ 


•J313m-Bip UI SSHDUt |C 




•pxjaH 


•9X3 ssojDtJ sanoui f^ '^loiqi ssqoui tj 




•ui3;S 


•59aj 01 X qoui f x ssqDui £ 



o 
en 



O 
en 



O 
en 



298 STEEL IN TENSION. [Art. 34. 

It will be observed that the process of annealing decreased 
both the ultimate resistance and elastic limit. The results are 
irregular, however, so far as the strains or elongations are con- 
cerned. 

Tables V. and VI. are taken from " Steel in Construction," 
a paper read before the Engineers' Society of Western Penn- 
sylvania, by Albert F. Hill, C. E., 20th April, 1880. Table V. 
contains the results of tests of specimen bars 3" X %" X 30". 
These specimens were cut from rolled bars which were subse- 
quently manufactured into eye-bars. 

Nine eye-bars containing 0.30 per cent, of carbon were 
made, besides nine others containing 0.50 per cent, of carbon. 
Each group of nine was divided into three classes, with welded, 
rolled and upset heads, respectively. These eighteen eye-bars 
were then put into the testing machine, and the results belong- 
ing to the nine containing 0.30 per cent, carbon are given in 
Table VI. 

The results of the tests of the bars containing 0.50 per 
cent, carbon, were not given, but it was stated by Mr. Hill 
that they verified the conclusions drawn from Table VI. 

The value of Table VI. is much enhanced by the fact that 
the results which it contains belong \.o full-sized bars, and not 
to prepared specimens. The bearing of the information which 
it gives on the mode of manufacture of the eye-bar head is 
also most important. It will be observed that the welded head 
possesses much less resistance than the others, and that the 
rolled head is a little stronger than the upset. 

Mr. Hill considered that these experiments " clearly estab- 
lish the fact that welding of high-grade steel for purposes of 
construction is out of the question in general practice." 

The metal in these bars was " open-hearth " steel made by 
Messrs. Anderson & Co., of Pittsburg, Penn. 

Table VII. gives the extension of the results on bars of 
Hay steel, "a part of which have already been given in the sec- 
tion on the coefficient of elasticity. The " elastic limit " is the 



Art. 34.] 



HA Y STEEL BARS. 



299 



intensity of stress at which "• traces " of permanent set begin 
to be observed, or immediately before. 

TABLE VII. 
Bars of Hay Steel, 



NO. 


LENGTH IN 
INCHES. 


SIZE OF BAR IN 
INCHES. 


POUNDS OF STRESS PER 
SQ IN. AT 


PER CENT. OF 




Elastic 
Limit. 


Ultimate 
Resistance. 


Final Stretch. 


Contraction. 


I 
2 

3 
4 
5 
6 


38.5 
18.7 


2.09 X I.I 

1.04 

5.20 X 1.34 
3.40 X 1.05 

0.08 
1.04 


54,000 
51,200 
49,200 
54,400 


93,200 
98,100 
93,900 
95,300 
108,100 
128,700 


10.00 
10.00 

8.00 
12.00 

6.00 




11.00 

7.00 

43.00 

35-00 

















Bar No. i was hammered down from a square bar 2.6 x 2.6 
inches ; and No. 4 likewise from a 6.24 x 1.56 inch bar. 

The extension, or stretch, in No. 3, was taken just before 
failure. 

Experiment No. 6 was made at a temperature of 22°(Cent.?) 
below zero. 

The '' Report on the Renewal of the Niagara Suspension 
Bridge," by Mr. Leffert L. Buck, C. E., 1880, has already been 
alluded to, in connection with the coefficient of elasticity. 
That " Report " contains the results of experiments made on 
plate specimens, of the Hay steel used in that work, with cross- 
sectional areas varying from about 0.32 square inch to 0.72 
square inch, and with lengths varying from seven to ten inches 
between clamps. 



300 STEEL IN TENSION. [Art. 34. 

These specimens were subjected to various kinds of treat- 
ment, such as punching, anneahng, blows while under stress, 
nicking on edges, etc., etc. 

One specimen was also subjected to intermittent stresses. 

These manipulations necessarily affected the elastic limit 
and ductility as well as the ultimate resistance. With the 
omission of the one specimen just mentioned, the extreme re- 
sults are as follows : 

FlactJp HmJ^ i Greatest 59»43i pounds per square inch. 

Ji^lastic limit -j Least 43,300 

Ultimate resistance. \ ^ . ^i\^^ 

I I-east 59.370 






Final stretch \ Greatest 19.4 per cent. 

rinai siretcn ] Least 7.0 " " 

Final contraction of j Greatest 42.0 " " 

ruptured section. . ( Least 13.0 " " 



The "stretch" and "contraction" are per cents of ten 
inches and original section, respectively. 

Table VIII. contains the results of the experiments of Sir 
Wm. Fairbairn on the different varieties of English steel given 
in the left-hand column. The specimens were one inch square, 
and had previously been subjected to a transverse load. The per 
cents of strain or elongation are for a length of eight inches, 
which, it is presumed, included the section of fracture. 

Table IX. contains the results of tensile experiments on 
Bessemer and crucible steel specimen bars by Mr. Kirkaldy for 
the " Steel Committee " (English). 

The first part of the table gives the results of experiments 
on bars turned accurately to a diameter of 1.382 inches with a 
length " in the clear " of 50 inches. It is presumed that the 
per cents of elongation apply to that length. 

The second part of the table gives the results of experi- 
ments on bars "in their natural skins," with a diameter of 1.5 
inches and length of 120 inches; to which length the per cents 
of elongation apply. 



Art. 34.] 



ENGLISH BAR SPECIMENS. 



301 



TABLE VIII. 
Bar Speci)nc7is. — 1867. 



PRODUCERS. 



Messrs. Br 01071 iSr' Co. 

Best cast steel, for turning tools 

Best cast steel, milder , 

Cast steel from Swedish iron for tools 

Cast steel, milder, for chisels 

Cast steel, mild, for welding 

Bessemer steel 

Double shear steel from Swedish iron, 

Foreign bar, tilted direct 

English tilted steel 



C. Canunel ^ Co. 

Cast steel, termed " Diamond Steel ". . . 

Cast steel, termed " Tool Steel " 

Cast steel, termed " Chisel Steel" 

Cast steel, termed " Double Shear Steel " 

Hard Bessemer steel 

Soft Bessemer steel 



Alessrs. Naylor, Vickers df Co 

Cast steel, called " Axle Steel " 

Cast steel, called " Tire Steel " . . . . 

" Vickers Cast Steel, Special " 

" Naylor & Vickers' Cast Steel " . . . 



S, Osborne. 

Best tool cast steel 

Best chisel cast steel 

Sates-cup, shear blades, etc 

Best cast steel for taps and dies .... 
Toughened cast steel for shafts, etc 

Best double shear steel 

Extra best tool cast steel 

Boiler plate, cast steel 



ULTIMATE RESISTANCE 


FINAL STRAIN OR 


PER SQ. INCH. 


ELONGATION. 


Pounds. 


Per cent. 


68,404 


25 


91.250 


1-5 


106,714 


I.O 


116,183 


3.62 


110,055 


3-31 


91,972 


19.62 


92,555 


5-43 


76,474 


13-56 


59,538 


21.06 



110,055 


1.53 


109,072 


1.50 


120,398 


2.50 


96,665 


2-37 


89,121 


20.87 


81,483 


20.43 



88,665 


16.25 


91,520 


9.00 


134,145 


1. 00 


118,066 


1-75 



98,942 


0.93 


123,686 


3.18 


115,849 


2.12 


98,790 


1.68 


103,116 


5-25 


87,931 


2.43 


85,724 


0.43 


111,676 


13-50 



302 



STEEL IN TENSION, 



[Art. 34. 



TABLE Ylll.— Continued. 



PRODUCERS. 



U. Bessemer. 



Hard Bessemer steel. . 
Milder Bessemer steel. 
Soft Bessemer steel . , . 



Sanderson Brothers. 

Cast steel from Russia iron for welding. 

Double shear steel 

Single shear steel 

Fagot steel, welded 

Drawn bar, not welded 



Messrs. Turton &f Sons. 



Steel for cups . 
drills 



cutters 

turning tools 
machinery. . . 

punches 

mint dies. . . . 
dies , 



taps 

Double shear steel. 



ULTIMATE RESISTANCE 
PER SQ. INCH. 



Pounds. 

103,085 

88,175 

78,606 



83,484 
107,940 

107,182 

75,199 
103,960 



100,155 

87,552 


2.75 
1.06 


95,372 
80,273 


1-37 
0.12 


102,915 
102,567 

106,237 

87,471 


1.43 
1.62 

2.87 
87 


97,994 
73,266 


1.87 
0.81 



FINAL STRAIN OR 
ELONGATION. 



Per cent. 

1.87 
20.00 
19.12 



2.25 

3-31 
2.81 

125 

3-43 



The " Area of Fracture Section " (Table IX.) is the/^r cc7it. 
of original sectional area, which, multiplied by that original 
area, will give the area of the fractured section. The per cent. 
of contraction will then be given by taking the difference be- 
tween 100 and the number expressing the ^^ Area of Fracture 
Section^ 

The experiments of which the results are given in Table 
IX., are those for which the coefificients of elasticity were com- 
puted in Eqs. (2), (3), (4) and (5). 



Art. 34.] 



ENGLISH BAR SPECIMENS. 



303 



TABLE IX. 
Bar Speci7nens. — 1868 and 1870. 



NUMBER AND KIND OF SAMPLES. 



V 



CO 



5, Hammered, tires .... 
5, Hammered, axles. . . . 
4, Hammered, rails,.. . . 

4, Rol'd tires, axles, rails 

5, Hammered, tires . . ^ 

4, Hammered, axles (^.-9 

I, Hammered, rails... 

1, Rolled, axles J u 

3, Chisel 

3, Samples 

Tires 

Rods 

2, Samples 

3, Gun-barrels 

Hammered 

Hammered 

2, Rods 

2, Rolled 

3, Faggoted '\ b 

3, Samples ! | 

2, Samples (In 

3, Tires and axles J CQ 



J3 



Pounds. 
52,200 
49,000 
48,000 
43,200 
46,200 
57,300 
44,000 
42,000 
58,240 
57,120 
58,240 
60, 500 
45,900 
37,600 
56,000 
44,800 
59,900 
45,900 
43,700 
44,800 
39,200 
37,000 



Pounds. 
78,600 
75,000 
74,500 
71,500 
79,500 
91,700 
85,400 
68,600 
118,200 
114,300 
97,400 
93,700 
90,800 
86,300 
83,000 



75,400 
67,100 
79,300 
76,600 
75,300 
75,400 



PER CENT. FINAL 
ELONGATION. 



II. I 

12. I 
12.8 

17-5 
9.17 
8.72 
2.96 

10.56 

5-3 
7-3 
4-7 
1 .1 
4.1 
8.0 
8.0 



0.9 

2.0 

II. I 

II. 9 

II-5 
13 -.6 



AREA OF FRAC- 
TURE SECTION. 



55 
51 

52 

65 
62 

72 
96 
89 

94 

80 

94 
100 

95 
95 
98 



98 
97 
55 
54 
80 

58 



Some of the earlier experiments of Mr. Kirkaldy (1861), on 
various English (also Krupp's) steels, gave the following ulti- 
mate resistances in pounds per square inch for specimens of 
small area of cross section : 

Highest (forged cast steel, reheated and cooled gradually). = 148,300 
Lowest (forged puddled steel) = 42,600 

Table X. contains the results of twelve experiments by Mr. 
Kirkaldy (** Experimental Enquiry, etc., of Fagersta Steel," 
1873) on 2-inch square hammered bars of Fagersta steel, turned 



304 



STEEL IN TENSION. 



[Art. 34. 



TABLE X. 

Fagersta Steel Bars. 



1.2 
1.2 
1.2 



0.9 
0.9 

0.9 



0.6 
0.6 
0.6 



0.3 
0.3 
0.3 



POUNDS OF STRESS PER SQ. INCH, AT 



Elastic Limit. 



62,400 
60, 200 
63,500 



en 
O 



> 
< 



. o 

63,600) c^ 
62,400 >• "^ 
63,200) II 

> 

< 



62,500 

53,200 
58,600 



> 
< 



44,200) 
41,500 y 
40,600 ) 



> 
< 



Ultimate Resist. 



81,952 
81,424 
92,224 



> 
< 



CO 

M 
O 

112,976 ) o 
109,952 V M 

96,912) II 

> 

< 



CO 

o 



101,232 \ g 

97,968 \ M 

108,696) II 



> 

< 



61,288 
63,120 

59,528 



M 
CO 



> 

< 



PER CENT. OF FINAL 



Contraction. 




4-97) 
8.39^ 
4-97) 



21. 
10 
II 



CO 

46) 2" 

> 

< 




Elongation. 







FRACTURE. 



Granular. 



Granular. 



Granular. 



Silky. 



to 1. 1 28 inches diameter, with a length between shoulders of 
nine diameters or 10.15 inches. 



Art. 34.] WHITWORTH'S COMPRESSED STEEL. 



305 



^^ Mark'' indicates the relative hardness, 1.2 being the 
hardest and 0.3 the softest. 

The per cents are of the original sectional area (/. ^., of one 
square inch) and of the original length, which, of course, in- 
cludes the section of failure. 

Sir Joseph Whitworth manufactures his compressed steel 
by subjecting the molten metal to an intensity of pressure of 
13,000 to 14,000 pounds per square inch, immediately after it 
is taken from the furnace. 

Table XI. contains the result of some tensile experiments 
on sortie specimens of this steel. Each specimen is turned to 
a diameter of 0.798 inch (0.5 square inch in normal section) for 
a length of two inches, for which the per cent, of final elonga- 
tion is expressed (see " Proc. Inst, of Mech. Engrs.," 1875). 
The specimens are thus seen to be so formed as to give very 
high results, both for ultimate resistance and elongation. 

TABLE XI. 

Whitworth's Coinpressed Steel. 



DISTINGUISHING COLORS FOR 


ULT. RESIST. LBS. 


PER CENT. FINAL 


REMARKS. 


GROUPS. 


PER SQUARE INCH. 


ELONGATION. 










Axles, boilers, cranks, 


Red, Nos. I, 2, 3 


89,600 


32.0 


propeller shafts, 






^ 


( rivets, etc. 


Blue, Nos. I.. 2, 3 


107,500 


24.0 


j Shafting, drill spin- 
{ dies, hammers, etc. 
( Large planing tools, 


Brown, Nos. i, 2, 3 


129,900 


17.0 


\ Large shears, drills, 

f etc. 

I Boring tools, finishing 


Yellow, Nos. I, 2, 3 


152,300 


10. 


tools for planing, 
etc. 


Special alloy with Tung- 








sten 


161,300 


14.0 





20 



3o6 



STEEL IN TENSION. 



[Art. 34. 



TABLE XII. 
Plates. — Unannealed. 



0.30 
0.30 
0.30 
0.30 
0.30 



0.40 
0.40 
0.40 
0.40 
0.40 



0.50 
0.50 
0.50 
0.50 
0.50 



t^ IS 









a 






TENSILE STRESS IN POUNDS PER SQUARE 
INCH AT 



Elastic Limit. 



43,260 
44,820 
45,110 

43,990 
44,720 



51,620 

50,980 
51.260 
51,100 
50,890 



58,950 
59,200 
58,540 
5S,88o 
59,330 



o 

CO 



< 



o 



> 



O 






Ultimate. 



79.120 

77,840 
78,390 
77,970 
78,280 



81,990 
81,720 
83,730 
81,830 
83,130 



85,790 
86,220 
85,560 
86,000 
86,330 



> 
< 



> 



> 
< 



H 




2 




Z 









ij 




h 


CHARACTER OF 




b. 


< 














cu 






a 


FRACTURE. 


> 

< 







19-3 



13-9 



10.5 



Fine and silky. 



Very fine. 



Good ; slightly 
granular on 
edges. 



Boiler Plate, 



Table XII. was also taken from Mr. Hill's paper, and con- 
tains results obtained from tests of large pieces of boiler plate 
of mild steel of the same character as that used in the bars the 
results of the tests of which are given in Table VII. The stress 
tvas in the direction of rolling. 



Art. 34,] 



BOILER PLATE. 



30/ 



TABLE XIII. 



Plates. — Unannealed. 



JJtTZ A "KT /^C 


DIMENSIONS OF 


STRESS IN POUNDS PER SQ. INCH. 


PER CENT. OF 





iVlilAIN yJr 


i5rlLCIMfc.N5> IN 






S 1 Kfc. H_ri OF 


CO 




INCHES. 


Elastic Limit. 


Ultimate. 


ORIg'l LENGTH. 


II II 

in 

■*■> 
C 
0) 


3 


if X 4 


47,680 


66,440 


19.6" 




3 


4 X i 


53,400 


68,900 


19.9 


t/5 


**-" ^ 

QJ 


3 


2 X 1 


39,670 


62,360 


26.3 






3 


•hk X i 


38,940 


62,920 


27.7 





(U 


4 


2f X i 


38,390 


64,860 


24.8 




M 




3 


If X i 


53,790 


69,560 


19.0 


't 

*£ 


3 


4x4 


44,050 


65,830 


26.3 


Ul 




,_, 


3 


2 X f 


40.160 


65,690 


23-9 


\M 


"rt ^ 


3 


3§ X 1 


41,690 


65,200 


21-5. 




.s - 


2 


I X k 


49,485 


77,270 


23.6 


tfl 


-l-J 


2 


I X i 


50,745 


76,925 


24.8 


4= 




2 


I X f 


42,985 


70,375 


29.8 





I-. 


2 


I X 1 


43,160 


70,530 


22.5 




-t-i >. 


9 


I X 4 


45,280 


64,970 


29. 1 


U 


'3 


7 


1X8 


41,810 


64,640 


28.6^ 








5 


2 X 1 


41.368 


67,130 






-4-J +J 




4 


2 X 1 


38,390 


66,990 






1-. (U 




13 

1 


I| X 1 


39,430 


65,700 


For 10 ins. 23.2 


H-I 



Table XIII. contains the mean results of Prof. Kennedy's 
experiments on mild boiler plate containing about 0.18 per 
cent, of carbon (London "Engineering," Vol. XXXI., 1881). 
The left column shows the number of tests in each group, for 
which the other columns contain the mean results. The dimen- 
sions of specimens are not exact, but closely approximate. The 
length for which the per cent, of elongation is given, in all 
cases, contained the fracture. Consequently those ''Per cefitsT 
include the "local" extension which exists at the section of 
fracture. This accounts for the larger values, as a rule, which 
are found for the 4-inch lengths; 21^-inch lengths, containing 
the section of fracture, gave much higher values. 



308 



STEEL IN TENSION. 



[Art. 34. 



The first five groups were pulled from pins, and the next 
four from wedge grips. The manner of holding the test 
pieces, however, was not observed to have any influence on 
the results. 

Within the limits of these experiments, also, the ratio of 
width to thickness of the specimen seemed to have no influ- 
ence. It will be observed that Prof. Kennedy's specimens 
were all (what may be called) '^ long " specimens. 

His experiments on some annealed specimens of this steel 
showed that the process of annealing reduced the ultimate 
resistance only 3 or 4 per cent. 

TABLE XIV. 



Fagersta Plate. 





* 


ELASTIC LIMIT IN LBS. PER SQUARE INCH. 


ULT. RESIST. IN LBS. PER SQUARE INCH. 


2 


Large. 


Small. 


Long. 


Mean. 


Large. 


Small. 


Long. 


Mean. 


5 


J. 

1 

1 

8 


53,300 
37,900 
29,500 
31,100 
28,000 


50,500 
35,400 
29,300 
30,800 
28,300 


38,900 
35,600 
25,400 
27,500 
26,100 


47,567 
36,300 
28,067 
29,800 
27,467 


74,915. 
60,480 

51,456 

55,803 

52,924 


71,940 
56,740 

50,345 
54,425 
52,475 


55,135 
54,140 

48,925 
50,160 
49,280 


67.330 
57,120 
50,243 

53,463 
51,560 


Mean. 


35,960 


34,860 


30,700 


33,840 


59,116 


57,185 


51,528 


55,943 


•a 


c 
c 

< 


C L 

8 
1 
4 

f 

i 

5. 

L 8 


35,500 
33,800 
28,900 
27,800 
25,500 


33,200 
30,500 
28,100 
27,900 
25,700 


26,700 
29,800 
25,900 
27,300 
25,200 


31,800 
31,367 
27,633 
27,667 

25,467 


57,485 
54,543 
51,076 
51,338 
50,432 


55,459 
52,715 
50,350 
50,842 
50,025 


45,460 
49.605 
46,740 
49,490 
47,455 


52,801 

52,288 

49,389 
50,557 
49.304 


Mean. 


30,300 


29,080 


26,980 


28,787 


52,975 


51,878 


47,750 


50,868 



Fractures all " silky." 



Art. 34.] 



BOILER PLATE. 



309 



TABLE XV. 

Fagersia Plate. 



i 

u 
z 


X 


PER CENT. CONTRACTION OF AREA. 


PER CENT. FINAL STRAIN OR STRETCH. 


s 


Large. 


Small. 


Long. 


Mean. 


Large. 


Small. 


Long. 


Mean. 


a; 

c 

;3 


1 
4 
a 
« 

ft 

L 8 


43.1 
4S.5 
59-3 
50.0 

55-1 


47.1 
54.2 
62.5 
58.6 
61.7 


37-9 
59-7 
71.0 
61.2 

60.7 


42.7 
54-1 
64- 3 
56 6 

59-2 


10.8 
28.2 
36.1 

36.4 

37-2 


13-5 
35-5 
41-5 
40.0 

44.7 


5. 21 
10. 17 
20.64 
16.30 
17-95 


9-45 
24.41 

32-57 
30.78 

33.01 


Mean. 


51-2 


56.8 


58. 1 


55-4 


29-7 


35-0 


14.05 


26.04 


•6 

c 


r 1 

X 

I 

4 

a 

8 
i 
3 

5. 

L 8 


57.1 
60.9 

63-4 
61.0 
62.0 


60.8 

63-5 
63.6 

65.1 

64-3 


64.6 

67-5 
69.6 

64-3 
63.1 


60.8 
64.0 
65.5 
63 5 
63.1 


22.9 
33-8 
35.8 
38.5 
34-4 


28.4 
40.1 
42.0 
42.5 
43-5 


10.98 
16.88 
18.19 
19-15 

17-45 


20.37 
29.99 
31.76 
33-08 
31-51 


Mean. 


60.9 


63.5 


65.8 


63-4 


33.1 


39-3 


16.50 


29.52 



In his paper Prof. Kennedy explains in detail his "■ elastic 
limit." It is the point at which the material "■ breaks down," 
and considerably above the elastic limit as analytically defined 
in this work. 

Tables XIV. and XV. exhibit the results of Mr. Kirkaldy's 
experiments (in the direction of rolling) on some Fagersta 

steel plate specimens. 



\i 



Plate Specimen 



1/ 



r 



Fig.l 



^ 



The plate from which 
these specimens were 
taken was marked 0.15, 
and the material was 
a mild steel. The 



^^ Large'' and ^^ Small'' specimens were shaped as shown ia 



3IO STKEL IN TEJsrsiON. [Art. 34. 

Fig. I. The width BC ox AD of the reduced portion was ten 
inches for the ^' Large " pieces, and one and one half inches 
for the ^^ Smair' ones. For the ^' Large'' specimens, the 
length of the reduced portion (AB or CD) was ten inches 
( = width), and four and one half inches ( = 3 widths) for the 
^^ Small y The ^^ Long'' specimens were 100 inches by 2^ 
inches '* in the clear." 

The results embodied in these two tables are of greater 
interest and value in consequence of the variety in the relative 
dimensions of the specimens. They show the important part 
played by " lateral strains " both in the ultimate resistance and 
final strains, or elongations, of test specimens. 

With very few exceptions the following general principle 
may be deduced from Table XIV. : 

Both the elastic limit and tiltimate resistance increase with 
the ratio of the width over the tJiickness of the plates. 

Nearly all the exceptions are in the results which belong to 
the Y^ unannealed, and the ^^ Long" annealed, specimens. It 
may be observed in connection with Table III., that the char- 
acter of the former specimen (possessing a low and irregular 
value of E) is decidedly abnormal, to which, undoubtedly, its 
exceptions are due. Annealing the long specimens seems to 
cause the disappearance of essentially all influence of the rela- 
tive dimensions of the cross section, where the ratio of width 
over thickness is, comparatively speaking, small. 

One origin of the results above stated is plainly to be 
found in the lack of lateral contraction in the plane of the 
plate, in accordance with the principles shown in Article 32, 
" Ultimate Resistance and Elastic Limit'' 

An examination of Table XV. shows the following general 
result, which, however, has more exceptions than the pre- 
ceding : 

TJie final contraction and elongation increase as the ratio of 
width over thickness decreases. 

With the long specimens, this does not seem to hold for 



Art. 34.] 



BOILER PLATE, 



311 



less values of the ratio than 2)^ -^ ^ = 6. Whether these 
principles may hold true, as general ones, or whether they may 
hold within certain limits (a possibility indicated in the *' Loftg'' 
specimens), the number and character of these experiments 
does not permit to be decided. They show, however, that the 
partial prevention of lateral strains in one direction, whatever 
may be the cause, will affect, to a considerable extent, experi- 
mental results ; also, that in testing plates the shape and rela- 
tive dimensions of the test piece should be carefully noted. 

TABLE XVI. 
Open-Hearth Steel Plates — 1880. 





PER CENT. 


LENGTHWISE. 


CROSSWISE. 












SPECIMEN, 
INCHES. 


OF 
CARBON. 


Stress in pounds per 
square inch. 


C 

§ 1 

u to 


Stress in pounds per 
square inch. 


C 










. 


I " 






Elas. Lira. 


Ult. Resist. 


pi 


Elas. Lim. 


Ult. Resist. 


^ 


f X i^ X 18 


0.30 


49,353 


93,339 


16 


49,510 


95,453 


18 


f X li X 15 


0.40 


63,227 


86,410 


14 


63,723 


87,780 


16 


-j-j XI X 12 


0.50 


65,070 


83,190 


10 


65,300 


84995 


15 



In Table XVI. are found the results of tests by Mr. Hill 
(" Engineers' Society of Western Pennsylvania," April 20th, 
1880), on specimens of open-hearth steel plate. Each result is 
a mean of three ^ and each specimen was cut from unannealed 
plate in a planer. It is to be particularly observed that each 
thickness of plate gave essentially the same elastic limit and ulti- 
mate resistance^ whether the direction of the testing stress was 
along or across the direction of rolling. 

Although the elastic limit increases with the amount of 



312 



STEEL IN TENSION. 



[Art. 34. 



carbon (consistently with the results in Table XII.), yet, it is 
very retnarkable to observe that the ultimate resistance decreases 
as the carbon increases, which is not consistent with the results 
contained in Table XII. 



TABLE XVII. 
Siemens Steel Plate — 1875. 







THICKNESS 

IN 

INCHES. 


POUNDS OF STRESS PER SQ. IN. AT 


PER CENT. 

FINAL 
STRETCH. 


PER CENT. 
FINAL CON- 
TRACTION. 


Elas. Limit. 


Ult. Resist. 



2 

U 


§1 

C rt 
^ C 


0.37 
0.71 


34,600 
30,400 


72,900 
66,900 


22.3 

24-5 


37-5 
44-7 


•0* 

C 
C 
<1 


0.37 
0.40 
0.40 
0.50 
0.62 
0.70 


31,500 
31,200 
29,800 
29,400 
26, 300 
24,500 


67,500 
66,400 
66,100 
65,800 
61,800 
60, 100 


24.8 

21 I 
24.8 
26.4 

25-5 
25.0 


43-1 
44-7 

38.5 
44-5 
43-3 
45-5 


Id 
to 

? 




C rt 


0.37 
0.71 


34,300 
30.400 


72,700 
67,300 


22.4 
24.7 


37-5 
43-6 


•a 

y 

C 
a 


0.37 
0.40 
0.42 
0.52 
0.62 
0.70 


31,200 
31,000 
30,000 
29,800 
26,300 
24,500 


66,900 
66,900 
65,800 
66,600 
60,600 
60,200 


26.4 
26.3 
20.4 
20.2 

22.7 
26.0 


46.6 
49.6 
39-0 
46.7 

35-3 
50.7 



The ratio of width over thickness of specimen increases 
from 2 (for the ^-inch, or, 0.30 per cent, carbon) to 5 ^^ (for 
the yV-i"ch, or, 0.50 per cent, carbon), and Mr. Hill considers 
this an explanation of this disagreement in the two sets of 
results. The results of a large number of tests on Fagersta 
steel specimens of considerable variety in the ratio of width 



Art. 34.] BOILER PLATE. S^S 

over thickness (Table XIV.) showed a regular increase, in 
both elastic limit and ultimate resistance, with an increased 
ratio of width over thickness. Agreeably to these results, 
therefore, the increase of carbon, in Mr. Hill's experiments, 
should have been accompanied by an increase in both elastic 
limit and ultimate resistance, since an increased ratio of width 
over thickness accompanied the increase of carbon. The dis- 
agreement seems inexplicable, but was probably due to the 
influence of some unnoticed peculiarity in the treatment of 
the material in the original plate, or of the specimens them- 
selves. 

Table XVII. contains the results of some specimen tests of 
Siemens steel plate, made by Mr. David Kirkaldy in 1875. 
The per cents of final stretch are for a length of eight inches, 
which contained the section of fracture. 

Tables XIII., XIV., and XVII. show that, as a general 
rule, both tJie elastic limit and ultimate resistance, in Tnild steel 
plates, increase as the thickness of the plate decreases. 

It is also seen that the process of annealing decreases both 
those quantities. 

Although Table XVII. shows no very marked result in 
regard to final stretch and contraction, yet when it is taken in 
connection with Table XV., it is clear that the process of 
annealing co7tsiderably increases both the final stretch and con- 
traction ; ' in other words, increases the ductility of the ma- 
terial. 

Again, Table XVII. shows that the ultimate resistance of 
steel plates is essentially the same, both in the direction of 
rolling and across it. This result is in agreement with that of 
Mr. Hill's experiments, as well as that of French experiments 
on Bessemer and Martin steel plates (Barba, on the *' Use of 
Steel," translated by A. L. Holley, pages 26 and 29). 



314 



STEEL IN TENSION. 



[Art. 34. 



Effects of Hardening and Tempering Steel Plates, 

In connection with the results given in Table XVII., Mr. 
.Kirkaldy found the following quantities by testing the same 
sized specimens of the same plates : 





Annealed. 




THICKNESS. 


ULTIMATE FINAL 
RESIST. STRETCH. 


FINAL 
CONTRACTION. 


0.64 inch. . . 
0.62 inch. . . 


. 57,100 pounds. ... 24. 1 per cent. . . 
. 60,500 pounds. .. . 20.2 per cent. . . 

Hardened. 


. 52.5 per cent 
. 48. 7 per cent 


0.64 inch. . . 
0.62 inch. . . 


. 64,700 pounds. .. . 22.4 per cent. . . 
. 65,050 pounds. .. . iS.o per cent. . . 


. 49.3 per cent 
. 45.5 per cent 



The hardening was done by heating to a cherry-red and 
cooling in water at a temperature of 82° Fahr. 



TABLE XVIII. 
specimen Tests. 



NOT TEMPERED. 


TEM?EREO. 


Pounds of Stress per sq. in. at 


Per cent. 


Pounds of stress per sq. in. at 


Per cent. 






Final 
Stretch. 




Final 
Stretch. 


Elas. Lim. 


Ult. Resist. 


Elas. Lim. 


Ult. Resist. 


58,'35o 


110,340 


13 


107,650 


169,430 


3-3 


56,800 


106,380 


15 


107,230 


163,320 


5-7 


55,720 


101,950 


17 


100,820 


153,370 


7.3 


53,178 


82,660 


19 


92,870 


140,580 


10.2 


50,850 


91,460 


21 


88,170 


129,250 


12.6 


47,980 


84,780 


23 


78,110 


116,460 


14.8 


45,160 


78.110 


25 


70,700 


104,810 


17.0 


42,020 


71,700 


27 


63.480 


93,430 


19-5 


39' 040 


66,300 


29 


56,800 


83,510 


22.0 


33,510 


58,640 


32 


46,860 


73,560 


24.2 



Art. 34.] RIVET STEEL. 315 

Table XVIII. gives the results of some experiments on a 
grade of French mild steel known as m^tal fondu. It is made 
by the Bessemer process, or in a Siemens-Martin furnace by 
" the substitution of ferro-manganese for spiegel, to produce 
carbonization." The table is a part of one given in A. L. 
Holley's translation of " The Use of Steel," by J. H. Barba. 
This steel was produced by the Siemens-Martin process, and 
the specimens were small ones, turned to a cross-sectional 
area of 0.31 square inch throughout a length of 3.93 inches. 

The tempering was done in oil at a bright red heat. 

It is thus seen that tempering or hardening increases both 
the elastic limit and ultimate resistance, but decreases both 
the final stretch and contraction of area. 

In these French experiments " it was observed that anneal- 
ing, well done, restored to the metal, in every case, its previous 
tenacity and elasticity." 

Rivet Steel, 

In Table XIX. will be found the results of the experiments 
of Prof. Alex. B. W. Kennedy (•" Engineering," 6th May, 
1 881). The steel was a very mild grade, for which coefficients 
of elasticity have already been given. 

The specimens were turned down, as shown, from J-i, {f 
and i^^g- inch " rounds." 

As in all other cases, the elastic limit and ultimate resist- 
ance are given per square inch of original section. 

Effect of Reduction of Sectional Area, i?t connection with 
Hammering and Rolling. 

Tables XX. and XXI. give the results of some of the ex- 
periments of Mr. Kirkaldy on Fagersta steel bars. The bars 
were originally three inches square, in normal cross section, 



3i6 



STEEL IN TENSION. 



[Art. 34. 



TABLE XIX. 

Rivet Steel. 



ORIGINAL DIA. OF 
BAR. 



Inch. 



16 

1 1 
It) 



A4 

IF 
-1-4 

16 



I-IV 



DIAMETER OF 
SPECIMEN. 



Inch. 

0.512 

0.505 
0.507 



0.616 
0.622 
0.616 



0.804 
0.804 
O.7S6 



POUNDS OF STRESS PER SQ. IN. AT 



Elas. Limit. 



o 



43,400) ^ 
45,200V ^ 
45,780) II 



46,370 

46,200 
46,220 



o 

o 



> 

< 



48,600^ 
47,730> 
46,750) 



o 

o 
o 



> 

< 



Ult. Resist. 



64,770) 
65,500 V 
65,770) 



> 
< 



67,960 j 

69,310 

69,210' 



o 

CO 

00 

o 



> 

< 



60,280) m" 
60,750 V '^ 
63,500) II 



> 

< 



PER CENT. OF 
FINAL STRETCH 
IN 10 INCHES. 



22.5) s 

^^5C II 
16.8 . 

> 

< 





and were hammered or rolled down to the dimensions shown 
in the second column in each table. Specimens were then 
turned down for testing to the diameters given in the third 
column, for a length of ten inches. The tables give results for 
duplicate specimens, one set having been unannealed and the 
other annealed. The fractures belonging to the 3x3 bars 
were all granular, and those belonging to the 0.5 x 0.5 bars 



Art. 34.] 



HAMMERING AND ROLLING. 



317 



TABLE XX. 

Fagersta Steel. — Uftajtnealed. 





BARS IN INCHES. 


DIA. OF 

SPECIMENS, 

INCHES. 


POUNDS OF 
SQUARE 


STRESS PER 
INCH AT 


PER CENT. 

FINAL 
STRETCH. 


PER CENT. 
FINAL 




Elas. Limit. 


Ult. Resist. 


CONT. 


Hammered . . 
Rolled 


0.5 X 
0.5 X 


0.5 
0-5 


0.357 
0.357 


78,300 

46,800 


95,960 
90,730 


6.9 
16.0 


47.0 
43.0 


Hammered. . 
Rolled 


I X 
I X 


I 
I 


0.619 
0.619 


49,800 
43,100 


83,720 
87,760 


16.0 
16.2 


44-7 
29-3 


Hammered. . 
Rolled 


1.5 X 

1.5 X 


1-5 
1-5 


1.009 
1.009 


46, 700 
40,500 


77,720 
79,280 


12.6 

10.2 


38.8 
15.8 


Hammered. . 
Rolled 


2 X 

2 X 


2 
2 


1.382 

1.382 


44,800 
38,300 


80,920 
84,073 


19.2 
15-9 


35-5 
20.8 


Hammered. . 
Rolled 


2.5 X 
2.5 X 


2.5 

2-5 


1.694 
1.694 


34,700 
36,600 


78,840 
72,585 


21.4 

8.2 


26.2 
10.3 


Hammered. . 
Rolled 


3 X 
3 X 


3 
3 


1.994 
1.994 


38,800 
30,400 


70,080 
62,393 


2.3 

2.5 


4.4 
4.4 



all silky ; the intermediate ones were partially silky and par- 
tially granular. 

As a part of the hammering and rolling was done at such a 

temperature as to essentially amount to cold hammering or 

cold rolling, the annealed specimens show more truly the 

effects of the two kinds of treatment than the others. 

The following results can be at once observed : 

The elastic limit y ultimate resistance and final contraction at 



3i8 



STEEL IN TENSION, 



[Art. 34. 



TABI.E XXI. 

Fagersta Steel. — Annealed. 





BARS IN mCHES. 


DIA. OF 

SPECIMENS, 

INCHES. 


POUNDS OF STRESS PER 
SQUARE INCH AT 


PER CENT. 

FINAL 
STRETCH. 


PER CENT. 
FINAL 




Elas. Limit. 


Ult. Resist. 


CONT. 


Hammered. . 
Rolled 


0.5 X 0.5 
0.5 X 0.5 


0.357 

0-357 


47,800 
41,200 


82,120 
80,210 


7-7 

9.8 


550 
51 


Hammered. . 
Rolled 


I X I 

I X I 


0.619 
0.619 


40,800 
40, 100 


78,650 
83,720 


15-2 

II-3 


54-0 
39-7 


Hammered. . 
Rolled 


1.5 X 1.5 
1.5 X 1.5 


1.009 
1.009 


42,300 
37,800 


77,810 
82,780 


13-7 

15-2 


47-7 

38.7 


Hammered. . 
Rolled 


2X2 
2X2 


1.382 
1.382 


41,300 
36,100 


78,893 
80,330 


17.7 
16.8 


41.2 
38.9 


Hammered. . 
Rolled 


2.5 X 2.5 
2.5 X 2.5 


1.694 
1.694 


31.300 
32,700 


66,140 
71,630 


14.7 
13.8 


45-4 

35-5 


Hammered. . 
Rolled 


3x3 
3x3 


1.994 
1.994 


29,800 
27,600 


69,640 
60,193 


7-7 
3.8 


8.4 
5-4 



section of fracture increase very much with the decrease of sec- 
tional area for either the hammered or rolled bars. 

Other and similar experiments verified these conclusions 
for both higher and milder Fagersta steels. 

The per cents, of final stretch are the greatest for the inter- 
mediate sectional areas, whether annealed or unannealed, while 
the relative effects of rolling and hammering are irregular. 

The hammered specimens invariably give the greatest final 
contraction, whether unannealed or annealed. 



Art. 34.] ' WIRE. 319 

If unannealed, the hammered specimens give the highest 
elastic limit and ultimate resistance ; if annealed, while this 
holds true (essentially) for the elastic limit, the rolled specimens 
give the highest ultimate resistance in four out of the six tests. 

Annealing decreases both the elastic limit and ultimate re- 
sistance ; this was also found to be the case for both higher and 
milder Fagersta steel specimens, which were similarly tested. 

In a set of 24 experiments (precisely the duplicates of 
those whose results are given in Tables XX. and XXI.) with a 
higher grade of steel, the greatest final stretch was found to 
belong to the smaller cross sections ; while in a similar set with 
a milder grade of metal, the greatest final stretch was found 
with the larger bars, whether the specimens were unannealed 
or annealed. 

Other relative effects of hammering and rolling were some- 
what irregular, and seemed to depend on the grade of steel. 

Effects of Annealing SteeL 

It has not been convenient to separately classify the experi- 
mental results showing the effects of annealing, but it has been 
seen that the process, in general, decreases both the elastic 
limit and ultimate resistance, and increases the ductility ; the 
lower grades of steel being the least influenced. 

Steel Wire, 

Table XXII. contains the results of testing, to ultimate 
resistance, the wire for which the coefficients of elasticity were 
given in Table I., together with some belonging to the Chrome 
Steel Co.'s wire, also tested by the engineers of the New York 
and Brooklyn bridge. The diameter of this wire was about 
0.165 inch (No. 8 Birmingham gauge). As will presently be 
shown, some of the material was cast steel and other Bessemer 
steel, all having been hardened and tempered. 



320 



STEEL IN TENSION. 



[Art. 34. 



TABLE XXII. 

Steel Wire. 



PRODUCER, 



J. Lloyd Haigh (i) 

Cleveland Rolling Mills (2) 

Washburn & Moen (3) 

Sulzbacher, Hymen, Wolff «& Co. .(4) 

John A. Roebling's Sons Co (5) 

Johnson & Nephew (6) 

Carey & Moen (7) 

Chrome Steel Co (8) 



ULTIMATE RESISTANCE IN 
POUNDS PER SQ, IN. 



Greatest. Mean. Least 



182,450 
182,576 
184,019 
179^833 
179,019 
206,170 
194,227 
170,150 



175,340 


166,169 


178,400 


172,984 


176,457 


169,706 


175,291 


167,807 


162,244 


125,321 


177,706 


163,027 


167,880 


126,814 


160,544 


150,657 



4.9 

1-7 
4-2 
2. 1 

4.5 



4.4 
1.8 
4.8 
0.4 
6.9 

3-1 
4.2 

0.5 
3-4 



K S 



O.161 
0.147 
0.161 
0.138 
0.147 

0.133 
0.162 
0.139 
0.167 
0.130 
O.T48 
0.129 
0.160 
0.125 



The column ^^ Per cent, final stretch** gives the highest 
values for the 5-feet lengths tested, and the lowest for the 100- 
feet lengths ; these were the greatest and least found. 

The column ^^ Dia. fracture** gives the greatest and least 
values of the diameter of the fractured section in decimals of 
an inch. There seemed to be no definite relation between the 
ultimate resistance and contraction of section of rupture. 

Col. W. A. Roebling states that the character of the above 
steel was believed to be as follows : 

(i) English crucible cast steel. 

(2) Open-hearth steel. 

(3) English crucible cast steel. 

(4) Krupp's Bessemer and cast steel. 

(5) Crucible cast steel and American Bessemer steel. 

(6) English crucible cast steel. 

(7) English crucible cast steel. 

(8) Crucible cast steel. 



Art. 34.] SHAPE STEEL. 32 1 

In Fairbairn's " Useful Information for Engineers," 3d 
series, p. 282, the following values are given for English steel 
wire (1866): 

ULT. STRETCH 
PRODUCER. DIA. ULT, RESIST. FOR 50 INS. 

Jenkins & Hill, soft patent steel . . . .0.085 in I05>730 lbs 0.53 

Jenkins & Hill, annealed steel 0.0S5 in 79,297 lbs 5 • 50 

Johnson 0.095 in 275,100 lbs 1. 71 

Johnson, patent steel 0.095 in 275,100 lbs 1 . 26 

The ultimate resistance is in pounds per square inch, and 
the final stretch in per cent, of original length of 50 inches. 

It is therefore seen that steel drawn into wire possesses an 
excess of resistance over that in larger masses, as bars ; it thus 
exhibits the same general phenomenon as wrought iron under 
similar circumstances. In fact the shape and dimensions of 
specimens have been seen to exhibit the same general effects 
on the results of testing as were found with wrought iron. 

Shape Steel. 

The following mean results, found by testing specimens of 
Bessemer and Martin Is and Ls, are given in A. L. Holley's 
" Use of Steel," by J. Barba : 

Unte7}ipcred Specimejis. 

ULT. RESIST. ULT. STRETCH. 

Bessemer Is 73, 500 lbs. per square inch 19-5 per cent. 

Bessemer Is 74,790 lbs. per square inch 21. i per cent. 

Martin Ls 64,960 lbs. per square inch 21.7 per cent. 

Martin Ls 67,210 lbs. per square inch 24.5 per cent. 

Tempered Specimens. 
Bessemer Is io6,.830 lbs. per square inch. . . . . 6.4 per cent. 

The tempering was done by heating to cherry-red and cool- 
21 



322 



STEEL IN TENSION. 



[Art. 34. 



ing in water at 50° Fahr. The dimensions of the specimens 



are not given. 



Steel Gun Wire. 



In 1875, W. E. Woodbridge, M.D., made a large number of 
tests on the mechanical properties of steel gun wires. The 
" wires " were about 0.3 inch square, having been drawn down 
from bars 0.375 inch square. The full, detailed account of 
these experiments is given in " Report on the Mechanical 
Properties of Steel, etc., by W. E. Woodbridge, M.D." 

The results given in this section are abstracted from the 
" Report " mentioned. 

TABLE XXIII. 

Gun Wires — Annealed. 



KIND AND MANUFACTURER. 



Crucible steel ; Hussey, Welles & Co (10) 

<C it < I t t t < 

( ( i( t i (( t ( 

(( << <( (( i( 

Martin steel ; N. J. Steel & Iron Co 

" " (10) 

(( (( t( <( <( t< 

It (( it H <( ti 

" Gun-screw wire " iron ; Trenton Iron Co. . . 

Chrome steel ; Chrome Steel Co (10) 

t i i< t i t i It 

<l t ( ^i H t I 

" " .'.".'.*.'.*. (10) 

Norway iron ; Messrs. Naylor & Co 

German steel ; Messrs. Park Bros. & Co .... 
Cemented cast steel ; Messrs. Park Bros. & Co. 



POUNDS OF STRESS PER 
SQ. IN. AT 



Elas. Lim. Ult. Resist. 



41,100 
26,800 
34,000 
39.700 
42,300 
37,500 
39,200 
49,000 
24,700 
43,400 
39,200 
43,200 
39,100 
26,000 
22,100 
35,100 
26,700 



92,300 
50,700 
61,700 
71,600 
72,100 
71,800 
71.500 
94,600 
52,600 
89,000 
77»700 
71,100 
89,100 
47,800 
5T,700 
61.700 
57,700 
74,900 



J 




< 




z 








b. 


X 







H 


H 


Z 


U 


u 


K 


u 


h 




(fl 


a. 




ii) 




&. 




5 


8 


22 


.0 


16 


.0 


15 


.0 


17 


• 3 


21 


■5 


18 


.0 


14 


I 


21 


I 


9 


I 


6 


3 


M 


2 


8 


7 


28 


5 


14 


2 


15 


7 


17 





19 


5 



45-0 
67.0 

59-0 
40.0 

46.0 
46.0 
46.0 
37-0 
57-0 
61.0 
61.0 
44.0 
41.0 
70.0 
42.0 
52.0 
50.0 
33-0 



Art. 34.] EFFECT OF TEMPERATURE. 323 

Table XXIII. gives results for wires which were annealed at 
bright red heat, without oxidation. 

The per cents, of final stretch are for five inches of original 
length, except in the case of specimens marked '*(io)," which 
indicates that the per cents, are for ten inches of original 
length. 

Other tests of wires about 0.3 inch square and iinannealed^ 
gave the following ultimate resistances in pounds per square 
inch of original section. The wires were of different varieties 
of steel, including cast and Martin steel. 

130,800. 84,400. 

106,900. 58,700. 

108,200. 59,200. 
135,000. 

The elastic limit varied from 35 to 92 per cent, of the ulti- 
mate resistance ; and the per cent, of final contraction varied 
from 1 1 to 43. The effect of annealing, both on resistance and 
ductility, is made very evident by comparing the two sets of 
results. 



Effect of Low and High Temperatures on Steel. 

The results of some German experiments and the expe- 
rience of the Massachusetts Railroad Commissioners with steel 
rails for one year, have already been given in connection with 
wrought iron. 

Table XXIV. contains the results of the experiments by 
Mr. Chas. Huston, as given in the " Journal of the Franklin 
Institute" for Feb., 1878. 

'* U. R." is the ultimate resistance in pounds per square 
inch, while " C." is the per cent, of contraction at the section 
of fracture. 

Each result is a mean of three experiments. 



324 



STEEL IN TENSION, 



[Art. 34. 



TABLE XXIV. 





" COLD." 


572'' FAHR. 


932° FAHR. 




U. R. 


C. 

26 

47 
36 

27 


U. R. 


C. 

23 
38 
30 

16 


U. R. 


C. 


ChErcoEl boiler-plate piled 


55,400 
54,600 
64,000 

78,400 


63,100 
66,100 
69,300 

82,800 


65,300 
64,400 
68,600 

77,300 


21 


Siemens-Martin (exceptionally soft). . . 

Crucible steel (ordinarily soft) 

Crucible steel (not quite hard enough 
to temper) 


34 
21 

20 







The method of producing rupture at the desired place was 
such as to make the specimens partake, to some extent at 
least, of the nature of " short " ones, which, however, would 
not affect the comparative results. 

It will be observed that the charcoal boiler-plate iron gave 
the highest resistance at the highest temperature, but that all 
the steels gave the highest '' U. R." at the intermediate tem- 
perature 572° Fahr. 

It is somewhat remarkable that in every case but the last 
(the hardest steel) the contraction of fractured section de- 
creased with the rise in temperature. 

Other results for steel will be found in Table IX. of Article 
35, and it will be seen that they tend to confirm the conclu- 
sions just drawn. 

In the " Annales des Fonts et Chaussees " for Feb., 1881, 
page 226, are given the number of breakages of steel rails 
which occurred in Russia in 1879. The following is the table 
showing the number of failures for each month of the year. 

These results conflict somewhat with those given by the 
Massachusetts Railroad Commissioners, in Art. 32. 



Art. 34.] CONSTRUCTIVE PROCESSES. 3^5 

January Ggg 

February 598 

March 854 

April 235 

May 235 

June 160 

July 247 

August 156 

September 214 

October 328 

November 341 

December 692 

The greatest number is found in the coldest half of the 
year, but the greatest number for any one month belongs to 
March, which is not the coldest month. It is probable that 
this is due to the effect of long wear on the frozen ground 
of the entire winter in connection with the possible alternate 
freezing and thawing of the ground in the month of March. 

Effect of Ma7tip Illations common to Constructive Processes ; also 
Punched, Drilled, and Reamed Holes. 

Table XXV. gives the results of the experiments of Mr. 
Hill (paper already cited in connection with Tables V. and 
VI.) on specimens of exactly the same size, and from the same 
steel plates, as those given in Table XVI. 

The different methods of preparing and treating the speci- 
mens are shown in the column headed '' Treatment of speci- 
men ^ 

With the exception of those of the lowest two 0.30 per 
cent, specimens, the results are averages of a number of ex- 
periments. 

From these results Mr. Hill concludes : 1st. " That both 
shearing and punching are injurious, per se, to all grades of 
steel, and cold punching far more so than shearing. 

2d. *' That both these operations afTect the elastic limit 
. . . far more than they do the ultimate resistance. 



326 



STEEL IN TENSION. 



[Art. 34. 



TABLE XXV. 

Open-Hearth Steet. 



PER CENT. 

OF 

CARBON. 



0.30 
0.30 
0.30 
0.30 
0.30 



0.40 
0.40 
0.40 
0.40 
0.40 



0.50. 
0.50 
0.50 
0.50 
0.50 



TREATMENT OF SPECIMEN. 



Cut in planer 

Sheared 

Punched 

Punched and hammered cold .... 
Punched, hammered and annealed 

Cut in planer , 

Sheared , 

Punched , 

Sheared and annealed , 

Punched and tempered 

Cut in planer 

Sheared 

Punched , 

Sheared and tempered , 

Punched and annealed , 



POUNDS OF STRESS PER 
SQUARE INCH AT 



Elas. Lim. 



49'43I 

32,370 

o 

o 

55,780 



63,475 

46,900 
o 

59-350 
52,780 



65,185 

51,666 
o 

60,375 
57,960 



Ult. Resist. 



94.396 
74,980 
63,410 
87,540 
100,410 



87.095 
75,330 
68,890 
86,160 
103,560 



84.092 
79.900 
78,400 

87.293 
84,900 



PER CENT. 
OF FINAL 
STRETCH. 



17.00 

10.00 

0.45 

0.55 

7-50 



15.00 
7.00 
5.00 

16.00 
7.00 



12.50 

5.00 

4.00 

17.00 

12.00 



3d. '■' That apparently the lower grades of steel are pro- 
portionately more injuriously affected than the higher 
grades. . . 

4th. "That the injurious effects of shearing and punching 
can be almost entirely counteracted by subsequent annealing, 
or tempering in oil from a low heat. 

5th. '' That annealing restores the elastic limit to a greater 
extent than the ultimate, while tempering as above, on the 
contrary, largely increases the ultimate resistance and ductility, 
but does not so fully restore the elastic limit." 

Table XXVI. shows the results of other experiments by 



Art. 34.] 



PUNCHING, DRILLING, ETC. 



327 



Mr. Hill, from the same paper, on the relative effects of drill- 
ing, punching and reaming, and punching (with and without 
annealing) rivet holes in steel plates. These plates were pre- 
cisely the same as those from which the results given in 
Tables XVI. and XXV. were obtained. The dimensions of 
the different specimens are given in the second column from 
the left. 

TABLE XXVI. 

Open-Hearth Steel, 



0.30 

0.30 

0.30 
0.30 



0.40 

0.40 

0.40 
0.40 



0.50 

0.50 

0.50 
0.50 



PLATE SPECIMEN. 



•K in. rolled plate, cut 
in planer on all edges. 
Strips lYz ins. wide, 18 
inches long. 



^ in. rolled plate, cut 
as above. Strips 1-4 ins. 
wide, 15 iiKhes long. 



"il; rolled plate, cut as 
above. Strips i inch 
wide 12 inches long. 



Drilled, i in. diameter 

Punched, 0.935 in. Ujameter 

Reamed to i.i in faiameter 

Punched and annealed, 0.935 i"- diameter 
Punched, 0.935 in. diameter 

Drilled, 0.6 in. diameter 

Punched, 0.5 in. I ^linmptpr 

Reamed to 0.62 in. f O'^^ieter 

Punched and annealed, 0.62 in. diameter 
Punched, c.62 in. diameter 

Drilled, 0.4 in. diameter 

Punched, 0.4 in. | diameter 

Reamed to 0.5 in. \ diameter 

Punched and annealed, 0.45 in. diameter. 
Punched, 0.45 in. diameter 



u > 


2; 






<; 
. 


J tL, 


§ ta 




J 


r u 


-^ 


ESIST 
M. OF 
ION. 


z 


« S [-. 


u 


• u 


u 


h O* Id 


V, 


J (A 00 


u 


D 


a, 


98,966 


22.0 


100,700 


20.0 


78,070 


21.0 


66,108 


3-3 


59747 


15.6 


104,253 


19.0 


87,910 


18.9 


80,550 


5-0 


86,963 


29.0 


89,043 


26.0 


84,951 


31.0 


82,330 


15.0 



The relative influences of the different operations of drill- 
ing, punching, etc., will be emphasized by comparing the re- 
sults in Table XXVI. with those given in Table XVI. 

The operation of punching is seen to considerably injure 
the material in the vicinity of the punched hole. In every 
case, the punched specimen gives very much less resistance 



328 STEEL IN TENSION. [Art. 34. 

than any other. It is further to be observed that the injurious 
effect of the punch is only partially removed by annealing. 

Mr. Hill draws the following conclusions : 

1st. ** That the * reamed ' hole is the strongest, and follow- 
ing in the order of strength come the * drilled,' the ' punched 
and annealed,' and, lastly, the * cold punched ' hole. This 
graduation is well defined in all three groups. That the 
reamed hole should be stronger than the drilled hole, I am 
unable to account for. 

2d. " That the injurious effect of punching is local, and can 
be entirely removed by enlarging the hole sufficiently with 
either drill or reamer. The amount of drilling or reamxing re- 
quired after punching varies with the thickness of the plate 
and grade of steel. 

3d. " That although annealing is in a measure beneficial in 
partially restoring strength and ductility to the punched plate, 
it will hardly be found available for bridge work ; for, if you 
attempt to anneal before riveting, the holes will not fit ; if after 
riveting, you create internal strains of which no account can be 
taken, and which may subsequently produce failure. More- 
over, with proper machinery, punching and reaming will be 
found much cheaper than * punching and annealing.' " 

In regard to the excess of resistance with the '' punched and 
reamed " hole as compared with the drilled, it may be remarked 
that in every case the hole, as reamed, was greater in diameter 
than the drilled hole. There was, consequently, less material 
to be ruptured in the former case than the latter. This 
diminution of cross section makes the reamed specimen a 
*' smaller " one than the other and intensifies the '' shortening " 
effect of the rivet hoje. Both these influences would increase 
the ultimate resistance, per square inch, of the reamed speci- 
men beyond the drilled one ; but whether they supply the 
explanation for the whole difference is an open question. 

It will be observed that the ultimate resistances for the 
drilled and reamed specimens, in Table XXVI., run consider- 



Art. 34.] PUNCHING, DRILLING, ETC. 3^9 

ably higher than the corresponding quantities in Table XVI., 
or, indeed, those in Table XXV. ; for which the explanation 
is simple and obvious. The specimens for Table XXVI. car- 
ried one rivet hole each ; and at this rivet hole failure took 
place. The effect of the hole in any specimen was the restric- 
tion of the contraction to its immediate vicinity, and the par- 
tial prevention of the latter, which reduced the specimen, to a 
great extent, to a " short " one. An increase of ultimate re- 
sistance was, consequently, to be expected. 

The decrease of ultimate resistance with the increase of 
carbon has already been remarked upon in connection with 
Table XVI. 

The experiments of Prof. Alex. B. W. Kennedy, on the 
effect of punching and drilling 
holes, in mild-steel boiler plate, j '^ \^_J' 

are well illustrated by Table j } 

XXVII., which is condensed \ 1 

from one given in London ) j 

"Engineering," 6th May, 1881. / I 

None of these plates were an- J ^ (\_ 

nealed, but all were drilled or Fig. 2 

punched as received. 

Within the limits of these experiments. Prof. Kennedy 
observes, neither the width of the test piece nor the different 
diameters of die, had any essential influence on the results. 

The injurious eft^ect of punching is shown by the fact that 
the punched specimens gave only 92 to 98 per cent, of the 
resistance of the drilled ones. 

It will be noticed that both the drilled and punched speci- 
mens gave higher resistances than the natural plate. This is 
due to the " shortening " and other influence {i. e., the disturb- 
ance of the lateral strains) of the rivet holes, as before ob- 
served, and explained in Art. 32, *' Ultimate Resistance and 
Elastic Limit, " 




330 



STEEL IN TENSION. 



[Art. 34. 



TABLE XXVII. 

Punched and Drilled Holes. 



HOLE, INCHES. 



Drilled 

Drilled 

\ punch, W die 
^ punch, \~. die 

Drilled 

Drilled 

8 punch, I die. 
\ punch, \% die 



DIAMETER OF HOLE, 
INS. 



0.940 

0.940 
0.912 — 0.876 
O.S92 — O.S7I 

0.926 

0.934 
0.998 — 0.890 

o 945 - 0.875 



TENACITY IN 
TONS, SQ. IN. 
NET SECTION. 



TENACITY COMPARED WITH 
THAT OF 



Nat. Plate. 



38 


12 


38 


22 


35 


04 


34 


44 


35 


39 


34 


90 


33 


91 


34 


38 



105 
108 
000 

025 

126 
III 

073 



1 .096 



Dril'd Plate. 



0.918 
0.902 



0.965 
0.978 



Each result is a mean of four, from plate specimens 2, 4, 6 and 8 inches wide. 
The pitch of rivet holes across the middle of specimens was 2 inches, 
and the width of each specimen was so chosen that each side passed 
through the centre of a rivet hole, as shown in Fig. 2. A " ton " is 2,240 
pounds. The two diameters of punched holes are for the two sides of 
the plate. 



A duplicate set of experiments on 32 specimens of a some- 
what softer steel boiler plate, gave essentially the same results 
(see " Engineering," 6th May, 1 881). 

By experimenting on mild Fagcrsta steel plates with the 
thicknesses yi, }^, }i, ^ and S/^ inch, Mr. Kirkaldy found the 
ratio of the resistance of drilled specimens over that of punched 
ones to vary from about i.i (for }/s, % and ^-^-inch specimens) 
to 1.5 (for Yq specimen) when unannealed, and to be about i.i 
for all the thicknesses when annealed. All the specimens were 
12.5 inches wide, with three rows of 0.77 inch holes, pitched 
2.5 inches apart, running across the specimens. The average 
resistance for square inch of net section was greater than that 



Art. 34.] PUNCHING, DRILLING, ETC. 331 

of the original plate for the drilled holes, but considerably less 
for the punched ones. 

Mr. Kirkaldy states, " the loss from punching is not con- 
stant, but varies with the thickness, and also with the hardness 
of the material." He also concluded that punching hardens 
the material in the vicinity of the punch, and that the effect of 
punching is counteracted *' to a considerable extent " by an- 
nealing. 

The results of Mr. Hill's experiments, as given in Table 
XXVI., show, for the thickness of plates there used, that by 
enlarging the diameter of the punched hole from o. i inch to 
0.165 inch, by reaming, the injurious effect of the punch is 
entirely removed. 

Experiments on French steel plates, produced by the Bes- 
semer and Martin processes {inetal fondii), confirm this result 
and form a basis for other conclusions, as follows (*' The Use 
of Steel," by J. Barba, A. L. HoUey, translator, p. 40) : 

*' 1st. That the effects of punching and shearing are essen- 
tially local and spread only over a very restricted region, less 
than 0.039 hich on the edges of the sheared or punched parts ; 

" 2d. That no cracks exist in this altered region ; 

" 3d. That tempering destroys the effects of shearing and 
punching by bringing the metal back to the state it would be 
in if drilling or planing had been substituted for punching or 
shearing ; 

'' 4th. That annealing alone or after tempering destroys, as 
tempering alone does, the alterations caused by shearing and 
punching." 

These conclusions relate to plates from 0.27 inch to 0.46 
inch thick. 

In first-class practice, holes in steel plates and shapes are 
frequently first punched and then reamed to a diameter 0.125 
inch greater. 

Experiments on some narrow specimens of steel plate seem 
to indicate that conical punching (the die 0.16 to 0.20 inch 



332 STEEL IN TENSION. [Art. 34. 

greater in diameter than the punch) injures the material less 
than cylindrical punching (with a clearance of perhaps Jg inch). 

In the working of steel plates and shapes, during ordinary 
constructive processes, all local pressure of great intensity, and 
hammering while cold or at a low temperature, tend to pro- 
duce internal strains of great intensity or other changes in 
molecular condition which cause the finished plate or shape to 
be liable to great brittleness and unlooked-for failure of a local 
character. 

For these reasons M. Barba gives the following directions 
in regard to the working of steel : 

*' 1st. Avoid any local pressure of whatever nature it may 
be ; 2d. If local pressures have been produced by blows of a 
hammer, the action of the punch, etc. (which may, as we have 
seen, cause ruptures), heat the piece to a cherry-red in a very 
regular manner and as much as possible in its entirety — the 
whole of it at once — and let it cool in the open air on a homo- 
geneous surface, which has all over equal conducting power. 
This simple reheating, which may be considered as annealing 
for plates and bars, on account of their slight thickness, restores 
to the worked metal its original qualities, even if it was in a 
very unstable state of equilibrium." 

If a large amount of working (such as bending or curving) 
of a single kind is to be done to a single piece, it is best, if 
possible, to heat to a cherry-red and do the work by stages, 
rather than all at once ; and then anneal after the working is 
completed. If the working is local and the heating irregular, 
it may be necessary to anneal once or more during the progress 
of the work. 

Local heating in the production of the ordinary steel eye- 
bar head, for example, frequently gives much trouble, unless 
resort is had to subsequent annealing. 

These difficulties in the working of steel are -found more 
pronounced in the higher grades, and much experience is still 
needed before they can be entirely overcome. 



Art. 34.] 



BA USCHINGER'S EXPERIMENTS. 



333 



On account of the homogeneous character of the metal, 
upsetting processes, as in riveting, etc., seem to injure the 
molecular condition of steel much less than that of iron. 



Baiischinger s Experiments on the Change of Elastic Limit and 

Coefficient of Elasticity. 

The details of these experiments are given in " Der Civil- 
ingenieur," Part 5, 1881. The manner of application of the 
tests, and remarks on the quantities, elastic limit, stretch limit, 
and final load, will be found by referring to page 262. The 

following is the notation : 

« 

E. L. = elastic limit in pounds per square inch. 
S.-L. = stretch limit in pounds per square inch. 

F. L. = final load in pounds per square inch. 

E. =. coefficient of elasticity in pounds per square 
inch. 

Bessemer Steel. 





IN ORIGINAL CONDI- 










TION. 


AFTER 69 HRS. 


AFTER 0.5 HR. 


AFTER 68 HRS. 


E. L. 


•25,970 


43,272 


8,760 


14,970 


S.-L. 


40,380 


51,920 


55,470 


71,850 


F. L. 


46,140 


57,690 


70,080 






E. 


29,848,000 


29,549,000 


29,009,000 


30,146,000 



A small specimen of this Bessemer steel, about an inch in 
diameter, gave an ultimate resistance of 75,800 pounds per 
square inch. 



334 STEEL IN TENSION. [Art. 34. 

The elastic limit rises twice after two long periods of rest, 
and falls in a very marked manner after the short rest of 0.5 
hour. 

The stretch limit rises steadily while the coefficient of elas- 
ticity falls twice and then rises above its original value. 

Prof. Bauschinger was the first to determine, in regard to 
Bessemer steel, that by stretching the metal beyond its elastic 
limit its elasticity is elevated, not only during the time of 
action of the load, but also during a longer period of rest, with- 
out load, of one or more days ; and that, in this manner, the 
elastic lim.it may exceed the load which caused the stretching. 
(Dingler's Journal, Band 224.) 

Fracture of Steel, 

The character of steel fractures has been incidentally 
noticed, in some cases, in the different tables. 

If the steel is low, or if some of the higher grades are 
thoroughly annealed, the fracture is fine and silky, provided 
the breakage is produced gradually. In other cases the fract- 
ure is partly granular and partly silky, or wholly granular. 

In any case a sudden breakage may produce a granular 
fracture. 

Effect of Chemical Composition, 

The ten sets of results given in Table XXVIII. are taken 
from a great number of similar ones established by the United 
States Test Board, " Ex. Doc. 23, House of Rep., 46th Con- 
gress, 2d Session." The physical phenomena developed in con- 
nection with a given chemical constitution may be observed at 
a glance. 

The amount of final contraction of fractured section may 
be accurately estimated by comparing the ultimate resistances 
of the original and final sections. 



Art. 34.] 



CHEMICAL COMPOSITION, 



335 









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33^ 



COPPER, TIN, ETC., IN TENSION, 



[Art. 






The specimens were circular in section and either 0.625 inch 
or about 0.8 inch in diameter, while all possessed a length of 6 
inches. 

Art. 35. — Copper, Tin and Zinc, and their Alloys. — Phosphor Bronze. 

Coefficients of Elasticity. 

Table I. gives the coefficients of elasticity {^E) of the various 
metals and their alloys, according to the various authorities. 
These coefficients were determined by experiments in tension, 
and E is given in pounds per square inch. 

TABLE I. 



Brass 

Tin , 

Zinc 

Gun Metal , . , 

Zinc 

Zinc. . 

Copper 

Copper 

Brass 

** Berlin Brass 

Gun Bronze . . 

Alloy 

Alloy 

Tobin's Alloy 
Copper 



AUTHORITY. 



Tredgold. 



Wertheim. 



Thurston. 



8,930,000 

4,608,000 

13,680,000 

9,873,000 

12,828,000 
12,420,000 
17,702,000 
14,958,000 
12,148,000 
13,192,000 

11,468,000 

13,514,000 

14,286,000 

4,545,000 

9,091,000 



REMARKS. 



Cast metal. 

<( (< 
Copper, 8 ; Tin, i. 

Ingot. 

Annealed. 

ZnCu2. 

ZnsCuiT. 

Copper, 0.90 ; Tin, o. 10 (nearly). 
Copper, 0.80; Zinc, 0.20, 
Copper, 0.625 ; Zinc, 0.375. 
Composition, below table. 
Cast metal. 



Tobin's alloy is a composition of copper, tin, and zinc, in 
the proportions (very nearly) of 58.2, 2.3, and 39.5, respectively. 
The value of E for this metal, and those for the two preceding 
and one following it, are calculated for small stresses and 
strains given by Prof. Thurston in the *^ Trans. Am. Soc. Civ. 
Engrs.," for Sept., 1881. 



Art. 35.] 



COEFFICIENTS OF ELASTICITY. 



337 





TABLE II. 






Cast Jin. 




/. 


E. 


/. 


E. 


1,950 


1,147,000 


3,200 


96,400 


2,360 


472,000 


4,000 


41,540 


2,580 


172,000 


Broke at 


4,200 lbs. 




TABLE III. 


• 




Cast Copper. 




p- 


E. 

1 


P- 


E. 


800 


10,000,000 


12,000 


18,750,000 


2,000 


9,091,000 


13,600 


8,193,000 


4,000 


9,091,000 


16,000 


2,235,000 


8,000 


14,815,000 


22,000 


137,000 



Broke at 29,200 lbs. 



The values of E (stress over strain) for different intensities 
of stress (pounds per square inch) for cast tin, cast copper, and 
Tobin's alloy, are given in Tables II., III. and IV. 

"/ '* is the intensity of stress in pounds per square inch, at 
which the ratio E exists. 

Each of these metals is seen to give a very irregular elastic 
behavior. 

Tables II., III. and IV. are computed from data given by 
Prof. Thur^on in the United States Report (page 425) and 
" Trans. Am. Soc. Civ. Engrs.," already cited. 



22 



338 



COPPER, TIN, ETC., IN TENSION. 



[Art. 35. 



TABLE IV. 

Tobm's Alloy. 



/. 


E. 


/. 


E. 


2,000 


4,545,000 j 

1 


18,000 


5,455,000 


4,000 


4,545,000 


24,000 


5,941,000 


6,000 


4,688,000 


30,000 


6,250,000 


8,000 


4,938.000 


40,000 


6,390,000 


10,000 


5,263,000 


50,000 


4,744,000 


T4,ooo 


5,110,000 


6c,ooo 


3,436,000 



Broke at 67,600 lbs. 

Ultmiate Resistance and Elastic Limit. 

Table V. is abstracted from the results of the experiments 
of Prof. Thurston as given in the *' Report of the U. S. Board 
Appointed to Test Iron, Steel and other Metals," and *' Trans. 
Am. Soc. of Civ. Engrs.," Sept. 1881. The composition of the 
various alloys was as given in the table, which also contains 
results for pure copper, tin and zinc. All the specimens were 
of cast metal. 

The mechanical properties of the copper-tin-zinc alloys 
have been very thoroughly investigated by Prof. Thurston 
(''Trans. Am. Soc. of Civ. Engrs.," Jan. and Sept., 1881). As 
results of his work he has found that the ultimate tensile re- 
sistance, in pounds per square inch, of " ordinary bronze, com- 
posed of copper and tin . . . ., as cast in the ordinary course of 
a brass founder's business," may be well represented by : 

T, = 30,000 -j- 1,000/ ; 

" where / isthe percentage of tin and not above 15 per cent." 



Art. 35.] 



ULTIMATE RESISTANCE. 



339 



TABLE V. 



PERCENTAGE OF 



Copper. 



100 
100 
100 

90 

80 

70 

62 

52 

39 
29 
21 
10 
00 

00 

00 

Gun 
90 

80 
62.5 
58.2 
100 
90.56 
81.91 
71.20 
60.94 

58.49 
49 . 66 
41.30 

32.94 
20.81 
10.30 
0.0 
70.0 
57.50 
45-0 
66.25 
58.22 
10.00 
60.00 
65.00 



Tin. 



00 

00 

00 

10 

20 

30 

38 

48 

61 

71 

79 

90 
100 
Queensl'd 
100 
Banca. 
100 
Bronze. 

10 

00 

00 

2.3 
0.0 
0.0 
0.0 
0.0 
0.0 
0,0 
0.0 
0.0 
0.0 
0.0 
0.0 
0.0 

8.75 
21.25 

23-75 
23-75 
2.30 
50.00 
10.00 
20.00 



Zinc. 



00 
00 
00 
00 
00 
00 
00 
00 
00 
00 
00 
00 
GO 

00 I 

00 

00 
20 

37.5 

39-5 
0.0 
9,42 
17.99 
28.54 
38.65 
41.10 
50. T4 
58.12 
66.23 
77 63 
88.88 

lOO.OO 

20.25 
21.25 
31.25 

10.00 

, 39 48 
40.00 
30.00 
15.00 



POUNDS STRESS PER SQ. INCH AT 



Elastic Limit. 



11,620 
11,000 
14,400 
15,740 



5,585 
688 

2,555 
2,820 



3,500 
1,670 



2,000 

TO, 000 



10,000 
9, 000 
16,470 
27,240 
16,890 

3,727 

1,774 
9,000 

14.450 

4,050 
18,000 (?) 

1,300 

2,196 

3,294 
30,000(1^ 

5,000 (?) 
21,780 (?) 



Ult. Resist. 



PER CENT. FINAL 



19,872 
12,760 
27,800 
26,860 
32,980 
5,585 

688 

2,555 
2,820 
1,648 

4,337 
6,450 
3,500 

2,760 

3,500 

31,000 
33,140 
48.760 
67,600 
29,200 



32,670 

30,510 

41,065 

50,450 

30,990 

3,727 

1,774 

9,000 

14,450 
5,400 

31,600 
1,300 
2,196 

3,294 
66,500 

9.300 
21,780 

3,765 



Stretch. 



0.05 

0.005 

0.065 

0.037 

0.004 



0.07 
0.36 



0.36 

4.6 

32.4 

31.0 

4.0 

7-5 



31-4 
29.2 

20.7 

10. 1 

5-0 



0.16 

0.39 
0.69 

0.36 




Contract'n. 



10.0 

8.0 

15-0 

13.5 
00.0 
00.0 
00.0 
00.0 
00.0 
00.0 
00.0 
15.0 
75 -o 

47.0 

86.0 



40.0 

29-5 

8.0 

16.0 

43-0 
38.0 
28.0 
17 o 
II-5 



0.0 
0.0 
0.0 
0.0 



7.0 
0.0 
0.0 



340 



COPPER, TIN, ETC., IN TENSION. 



[Art. 35. 



TABLE Y. — Continued. 



PERCENTAGE OF 


POUNDS STRESS PER SQ. INCH AT 


PER CENT 


FINAL 


Copper. 


Tin. 


Zinc. 


Elastic Limit. 


Ult. Resist. 


Stretch. 


Contract'n. 


70.00 


10.00 


20.00 


24,000 (?) 


33, T40 


31 






75.00 


5.00 


20.00 


12,000 (?) 


34,960 


3-2 


5 4 


80.00 


10.00 


10.00 


12,000 (?) 


32,830 


I 


6 


4.0 


55 00 


0. 50 


44-50 


22,000 


68,900 


9 


4 


25.0 


60.00 


2. 50 


37-50 


22,000 


57,400 


4 


9 . 


6.6 


72.50 


7- 50 


2.00 


11,000 


32,700 


3 


7 


II .0 


77.50 


12.5 


10.00 


20,000 


36,000 





7 


0.0 


85.00 


12. 5 


2.50 


12,000 (?) 


34,500 


I 


3 


3-0 



The values of the elastic limit in the lower part cf the table wei-e not at all well 
defined. 



" For brass (copper and zinc) the tenacity may be taken 



as : 



T^ = 30,000 + Soo-s-. 

where ^ is the percentage of zinc and not above 50 per cent." 

He found that a large portion of the copper-tin-zinc alloys 
is worthless to the engineer, while the other, or valuable por- 
tion, may be considered to possess a tenacity, in pounds per 
square inch, well represented by combining the above formulae 
as follows : 

T^^ = 30,000 + 1,000/ -f 500^. 



These formulae are not intended to be exact, but to give 
safe results for ordinary use ^vithin the limits of the circum- 
stances on which they are based. 

Prof. Thurston found the ** strongest of the bronzes " to be 
composed of : 



Art. 35.] GUN METAL. 34 1 

Copper 55 -o 

Tin 0.5 

Zinc 44 . 5 

100. o 

This alloy possessed an ultimate tensile resistance of 68,900 
pounds per square inch of original section, an elongation of 47 
to 51 per cent, and a final contraction of fractured section of 
47 to 52 per cent. 

The first and sixth alloys of copper, tin and zinc, in Table 
v., are called by Prof. Thurston '' Tobin's alloy." ''This 
alloy, like the maximum metal, was capable of being forged 
or rolled at a low red heat or worked cold. Rolled hot, its 
tenacity rose to 79,000 pounds, and when moderately and care- 
fully rolled, to 104,000 pounds. It could be bent double either 
hot or cold, and was found to make excellent bolts and nuts." 

As just indicated for the particular case of the Tobin alloy, 
the manner of treating and working these alloys exerts great 
influence on the tenacity and ductility. 

Baudrimont found for a copper wire 0.0177 inch in diameter, 
an ultimate resistance of about 45,000 pounds per square inch? 
the wire being unannealed, while for a diameter of 0.064 inch, 
Kirkaldy found about 63,000 pounds per square inch. 

Prof. Thurston states : " brass, containing copper 62 to 70, 
zinc 38 to 30, attains a strength in the wire mill of 90,000 
pounds per square inch, and sometimes of 100,000 pounds." 

All of Prof. Thurston's specimens were what may be called 
''long " ones, i. c, they were turned down to a diameter of 0.798 
inch for a length of five inches, giving an area of cross section 
of 0.5 square inch. 

Gun Metal. 

Major Wade (" Reports of Experiments on Metals for Can- 
non," 1856) made many experiments on a gun metal composed 
of copper 89 and tin 11 (very nearly), called gun bronze. 



342 



COPPER, TIN, ETC., IN TENSION. 



[Art. 35. 



He found that different methods of manipulation of the 
molten metal and of treatment, as in cooling, affected to a great 
extent its resistance. 

TABLE VI. 
Guii Bronze. 



MINUTES IN 
LADLE. 



O 

29 



Highest. 
Mean, . , 
Lowest. 



ULTIMATE TENSILE RESISTANCE, POUNDS PER SQUARE INCH. 



Gun-heads. 



Small bars. 



17,698 17,825 

29,216 28,775 

23,381 24,064 



17,761 

28,995 

23.722 



50,973 31,132 

52,330 28,153 

56,786 j 28,082 



Density varied from 7.978 to 8.823. 

Table VI. gives the average results of a large number of 
experiments made by Major Wade. It shows the great range 
in the tenacity of the different specimens. 

General Results, 

Table VII. gives general results of various European ex- 
perimenters. T represents the ultimate tensile resistance in 
pounds per square inch. 

Some of these results are from the experiments of early in- 
vestigators, who attached little importance to the size and form 
of the test specimen. In all the cases the results would be 
more valuable if the circumstances of testing were given. 
Those belonging to the more unusual alloys, however, possess 
considerable general interest in spite of the uncertainty sur- 
rounding their experimental origin. The presence of a little 
phosphorus in copper is seen to increase its resistance in a 
marked manner. 



Art. 35.] 



VARIOUS ALLOYS. 



343 



TABLE VII. 




Copper, wrought 

Copper, cast 

Copper, bolts, with phosphorus o.oi 

Copper, bolts, with phosphorus 0.015 

Copper, bolts, with phosphorus 0.02 

Copper, bolts, with phosphorus 0.03 

Copper, bolts, with phosphorus 0.04 

/—Proportions.—. 

Gun metal, copper 12, tin i 

Gun metal, copper ii, tin i 

Gun metal, copper 10, tin i 

Gun metal, copper g, tin i 

, Weights in 100. > 

Alloy, copper 84 . 29, tin 15.71 

Alloy, copper 82.81, tin 17. 19 

Alloy, copper 81. 10, tin 18.90 

Alloy, copper 78 . 97, tin 2 1 . 03, brasses 

Alloy, copper 34.92, tin 65.08, small bells. . 
Alloy, copper 15.17, tin 84.83, speculum metal 
Tin 

/ Proportion . , 

Aluminium bronze, copper 90, Al. i 

Aluminium bronze, greatest 

Tin, cast 

Zinc, cast 

Brass, yellow 

Brass, yellow, copper 67, zinc 33 

Brass, tube, copper 62, zinc 38 

Brass, tube, copper 70, zinc 30 

Brass, wire 

Muntz metal, copper 60, zinc 40 

Sterro-metal, copper 10, iron 10, zinc 80 

Sterro-metal, copper 60, iron 3, zinc 39, tin 1.5 
Sterro-metal, copper 60, iron 4, zinc 44, tin 2.0 

— cast in sand ... 

— cast in iron, annealed 

— cast in iron, forged red hot 

Copper 60, iron 2, zinc 37, tin I 

Copper 60, iron 2, zinc 35, tin 2 

Copper 55, iron 1.77, zinc 42.36, tin 0.83 — 

cast 

— forged red hot 

— drawn cold 



Anderson. 



Mallet. 



Anderson, 

Rennie. 
Stoney. 
Rennie. 
Anderson. 
Everitt. 

Dufour. 
Anderson. 



33,600 
19,000 to 26,100 
16,900 
38,400 
45,400 
47,900 
50,000 

29,000 
30,700 
33,000 
38,100 

36, 100 
34.050 
39'650 
30,500 

3,140 

6.950 

5,600 

73,000 
96,300 

4.740 

3,000 
18,000 
28,900 
103,000 
80,600 
91,300 
49,300 

7,100 
53,8co 

43,100 
54,300 
69,400 
76,200 
85,100 

60,500 
76,200 
85,100 



344 



PHOSPHOR BRONZE IN TENSION. [Art. 35. 



Phosphor Bronze^ and Brass and Copper Wire, 

Table VIII. contains the results of the experiments of Mr. 
Kirkaldy on phosphor bronze, with two results each for brass 
and copper wire. 

TABLE VIII. 
Phosphor Bronze. 



METAL. 


E.L. 


ULT. RESIST. 
SQUARI 

Unannealed. 


POUNDS PER 
: INCH. 

Annealed. 


» 

FINAL STRETCH. 




55,800 
55,200 
40.500 
26,300 
21,700 


75,000 

74,000 

63,700. 

54>lOO 

50, 100 

102,750 

121,000 

121,000 

139,100 

159 500 

151,100 

63.100 

81,200 








\ a 
































' ' * wire 


49,400 
47,800 
53,400 
54,200 
58,900 
64,600 
37.000 
51,500 


37-5 
34-1 
42.4 
44.9 
46 6 
42.8 
34-1 
36.5 




( ( ( M 




( t ( ( ( 




( ( ( < ( 










( t ( ( ( 




Copp 
Brass 


)er wire 




, wire 









The diameter of the phosphor bronze wire varied from 
about 0.06 inch to o.i i inch ; that of the copper wire was 0.064 
inch, and that of the brass wire 0.0605 inch. 

The final stretch is the per cent, of the original length, and 
belongs to the annealed wire. 

The contraction of fractured section for the phosphor 
bronze specimens varied from about four to thirty-two per cent, 
of original area. 



Art. 35.] EFFECT OF HIGH TEMPERATURE. 345 

The first five results belong to metal of the same composi- 
tion but subjected to different treatment. 

Some specimens tested by Mr. Kirkaldy gave as low as 
about 21,700 pounds per square inch. 



Experiments on Rolled Copper by the " Franklin Institute 

Co7nniitteeJ" 

The results of these experiments are contained in the 
"Journal of the Franklin Institute," for 1837. 

That committee found, as a mean of 66 experiments, the 
ultimate resistance of rolled copper to be 32,826 pounds per 
square inch. The temperature of the copper varied from 62° 
to 82° Fahr. ^' The irregularities of strength in the different 
specimens varied from 1.9 to 4.8 per cent, of the mean te- 
nacity." 

The resistance was found to be the greatest at ordinary 
temperatures, and to decrease with acceleration as the tempera- 
ture increased. 



Variatio7t of Ultimate Resistance and Stretch at High 

Temperatures. 

The results contained in Table IX. were obtained at Ports- 
mouth (England) Dockyard, and were published in the Engi- 
neer, 5th Oct., 1877. ** i? " is the ultimate tensile resistance in 
pounds per square inch, and " St'' is the per cent, of stretch 
for a length of 10 inches in all except the last (steel) specimen. 

At 250° to 350° the gun-metal specimens lose about half 
their ultimate resistance and nearly all their ductility. Phos- 
phor bronze loses about one-third of its resistance and two- 
thirds of its ductility at 300° to 400°. Muntz metal and copper 
are not much affected, nor is cast iron. Wrought iron and 
steel gain in ultimate resistance but lose in ductility. These 



34^ 



ALLOYS LJV. TENSLON. 



[Art. 35, 



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Art. 35.] EFFECT OF HIGH TEMPERATURE. 



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34^ COPPER, TIN, ETC., IN TENSION. [Art. 35. 

results would probably be somewhat varied by different pro- 
cesses of, and treatment in, manufacture and construction. 
The Muntz metal and copper specimens were rolled. 



Bauschinger s' Experiments with Copper and Red Brass, 

Prof. Bauschinger extended his experiments on the repeated 
application of stress so as to cover not only wrought iron and 
steel, the results of which have already been given, but also 
copper and red brass. 

The notation is that already used : 

E, L. = elastic limit in pounds per square inch. 
S.-L. = stretch limit in pounds per square inch. 
E, L. = final load in pounds per square inch. 

E, = coefficient of elasticity in pounds per sq. inch, 
'liny = *' immediately." 

The copper specimens were of rolled material about 16 
inches long with a cross section about 2.4 inches by 0.64 inch. 
These specimens gave an ultimate tensile resistance, per square 
inch, of 28,900 to 32,000 pounds and a final contraction of 27 
to 46 per cent. 

The red brass specimens were turned to about one inch in 
diameter and 16 inches long. They gave ultimate tensile re- 
sistances, in pounds per square inch, varying from 19,600 to 
23,460. 

With one exception, in the second case of red brass, the 
elastic limit and stretch limit were elevated by repeated ap- 
plication of stress, whether immediately or at the end of fol- 
lowing periods of rest. 

The effects on the coeflficient of elasticity are seen to be 
somewhat irregular. 



Art. 35.] BAUSCHINGER'S EXPERIMENTS. 



349 



Copper. 





IN ORIGINAL CONDI- 
TION. 


im'y. 


im'y. 


im'y. 


E. L 
S.-L. 

F. L. 
E. 


5,475 


8,030 
11,670 

14,600 
17,249,000 


8,790 
14,650 
21,970 

16,154,000 


11,450 
22,880 




16,651,000 


15,770,000 



Copper. 





IN ORIGINAL CONDI- 










TION. 


AFTER 18 HRS. 


AFTER 23 HRS. 


AFTER 24 HRS. 


E. L. 


2,560 


7,320 


8,080 


11,520 


S.-L. 






14,680 


23,040 






F. L. 




14,650 


22,010 








E. 


16,011,000 


16,295,000 


15,197,000 


15,756,000 



Copper. 





IN ORIGINAL CONDI- 
TION. 


AFTER 43 HRS. 


AFTER 44.5 HRS. 


AFTER 51.5 HRS. 


E. L. 
S.-L. 

F. L. 
E. 


5,840 


8,030 

11,670 

14,600 

16,780,000 


10,340 

14,760 

22,160 

16,069,000 


15.390 
23,080 




16,097,000 


15,472,000 



350 



COPPER, TIN, ETC, IN TENSION [Art. 35, 



Red Brass. 





IN ORIGINAL CONDI- 










TION. 


im'y. 


im'y. 




E. L. 


7,680 


9,090 


9,260 






S.-L. 


13,960 


16,070 


19,240 




F. L. 


16,770 


19.550 










E. 


12,030,000 


12,485,000 


12,727,000 







Red Brass. 



E. L. 
S.-L. 

F. L. 
E. 



IN ORIGINAL CONDITION. 



5,600 
14,020 

16,820 
12,322,000 



AFTER 17.5 HRS. 



9.II5 

16,130 

19,640 

12,314,000 



AFTER 21 HRS. 



8,550 
19,240 



12,485,000 



Red Brass. 



E. L. 
S.-L. 

F. L. 
E. 



IN ORIGINAL CONDITION. 



3,480 

13,910 

16,690 

13,239,000 



AFTER 53 HRS. 



9,090 
16,070 



12,940,000 



Art. 36.] 



COEFFICIENTS OF ELASTICITY. 



351 



The explanation of the method of applying these repeated 
stresses will be found in connection with the results for 
wrought iron on page 262. 

Art. 36. — Various Metals and Glass. 

Coefficients of Elasticity. 

The following values of the coefficients of elasticity, in 
pounds per square inch, contained in Table I. are taken from 
Wertheim's *' Physique M^caniqiie,'' pages 57 and 58. The co- 

TABLE I. 



METAL. 


EXPERIMENTER. 


COEFFICIENT OF ELASTICITY. 


Drawn. 


Annealed. 


Lead 


We^theim. 


2,564,OCX) 

7,713,000 
11,564,000 
10,463,000 
16,721,000 
24,237,000 


2,457,000 

7,555,000 

7,942,000 

10,155,000 

13,920,000 

22,067,000 


Cadmium 


Gold 


Silver 


Palladium 


Platinum 





efficients are the means of a large number of tensile experi- 
ments, with the exception of that for cadmium, which was 
derived from experiments on transverse vibrations. This last 
method gave results which differed, in most cases, from the 
direct tensile ones not more than the latter did from each 
other. 

Wertheim also gives for the tensile coefficients of elasticity 
of some different glasses : 



352 METALS AND GLASS IN TENSION. [Art. 36. 

Mirror glass E ^= 8,792,000 pounds per square inch. 

Goblet (common) E = 9,559,000 " " " *' 

Goblet (fine) ^ = 8,589,000 " 

Goblet (violet) ^=7,110,000 " 

"Crystal" ^ = 5,830,000 " 

Ultimate Resistance and Elastic Limit, 

Wertheim determined the elastic limit of many of the 
more rare metals, such as those named in Table I., and they 
are here given in pounds per square inch : 

ANNEALED. DRAWN. 

Lead 2S4 to 355 

Cadmium 142 to 171 

Gold..... 4,266 to 19,200 

Silver 4,266 to 16,350 

Palladium 7, no to 25,600 

Platinum 20,600 to 37, 000 

His " limit of elasticity ** is that fofce which will perma- 
nently elongate the metal 0.000,05 of its original length, and 
all his experiments were made on wires of very small di- 
ameters. 

The following ultimate resistances were found for wires 
about -g-Vth Inch in diameter by Baudrimont (" Annales de 
Chlmie," 1850) : 

Gold 17,100 to 26,200 pounds per square inch. 

Silver 40,300 to 40,550 " " " " 

Platinum 32,300 to 32,700 " " " " 

Palladium 51,750 to 52,640 ** " " " 

The ultimate resistances of some other metals are : 

METAL. EXPERIMENTER. ULT. RESIST. 

Cast lead Rennie 1,824 pounds per square inch. 

Sheet lead Navier 1,926 " 

Pipe lead Jardine 2,240 " 

Soft solder (5- tin, \ lead). Rankine 7,500 *• " " 



Art. 37.] CEMENT AND BRICK, 353 

Sir Wm. Fairbairn f' Useful Information for Engineers/' 
second series, pages 226 and 267) found the following ultimate 
resistances in pounds per square inch by direct pull on straight 
tensile specimens : 

Flint glass 2,413 pounds. 

Green glass 2,896 " 

Crown glass 2,546 '* 

The specimens were of circular section and about 0.53 inch 
in diameter. 

By subjecting spherical glass shells to internal pressure he 
found the following ultimate resistances In pounds per square 
inch : 

Flint glass 4, 200 pounds. 

Green glass 4, 800 ' ' 

Crown glass 6,000 " 

The thickness of these shells varied from about 0.02 (crown 
and green glass) to 0.08 (flint glass) inch. 

Art. 37. — Cement, Cement Mortars, etc. — Brick. 

The ultimate tensile resistance of these materials depends 
upon many circumstances, and only a few out of a great num- 
ber of experimental results will be given. These results will 
be so chosen as to be representative, but a full and detailed 
knowledge of the action of cements and cement mortars, under 
different circumstances of testing and variety of composition, 
must be acquired by an examination of the original memoirs. 

Mr. Bremermann, during the construction of the St. Louis 
bridge, found in 18 experiments with pure " Fall City" (Louis- 
ville) cement : Elastic limit, 16 to 104 pounds per square inch, 
with a mean of 72 ; ultimate resistance, 35 to 147 pounds per 
square inch, with a mean of no; coefficient of elasticity, 
800,000 to 6,930,000 pounds per square Inch, with a mean of 

2,239,000. 

23 



354 



CEMENT AND BRICK IN TENSION. [Art. 37. 



TABLE I. 



KIND OF CEMENT. 



Toepffer, Grawitz & Co., Stettin, Germany 

Hollick & Co. , London 

Wouldham Cement Co., London % 

Say lor' s Portland Cement, Coplay, Pennsylvania 

Wampun Cement & Lime Co., Newcastle, Pennsylvania. . . 

Pavin de Lafarge, Teil, France 

A. H. Layers, London 

Francis & Co 

Wm. McKay, Ottawa, Canada 

Borst & Rog-genkamp, Delfzyl, Netherlands 

Louqudty & Co., Boulogne-sur-mer, France 

Riga Cement Co., Riga, Russia 

Scanian Cement Co., Lomma, Sweden 

Bruno Hofmark, Port-Kund. Esthland, Russia . . 

Coplay Hydraulic Cement, Coplay Cement Co., Coplaj% Pa 

Charles Tremain, Manlius, N. Y . . 

Allen Cement Co., Siegfried's Bridge, Pennsylvania 

P. Gouvreau, Quebec, Canada 

Riga Cement Co. , Rig^a, Russia 

Anchor Cement Co., Coplay, Pa 

Cumberland Hydraulic Cement Co., Cumberland., Md 

Society Anonyme, France 

Howe's Cave Cement, No. i, Howe's Cave, N. Y 

N0.2, '' 

No. 3, " 

Society Anonima, Emilia, Italy, ist quality 

'■ "• " " 2d quality 

Thomas Gourdy, Limehouse, Ontario, Canada 

A. H . La vers, London 

Scott's Selenitic Cement (Howe's Cave lime and plaster). . . 

Parian Cement, Francis «& Co., London, ist quality 

" '' "■ " " 2d quality 

A. H.Lavers, " 



NO. 


C. 


NO. 


TESTS. 




TESTS. 


12 


1.4.39 


3 


10 


1.330 


3 


12 


1,140 


3 


8 


1,078 




12 


568 


3 


12 


931 


3 


6 


926 




14 


go7 


3 


10 


882 


3 


12 


826 


3 


12 


764 


3 


5 


603 


2 


14 


606 


3 


6 


580 


2 


8 


292 


2 


12 


276 


3 


12 


276 


3 


8 


234 


2 


6 


230 


2 


12 


208 


3 


12 


196 


3 


7 


184 


2 


6 


123 


2 


10 


170 


2 


8 


T70 


2 


12 


181 


3 


12 


154 


3 


8 


126 


2 


6 


122 


2 


20 


2g8 


5 


8 


1,175- 


3 


12 


C96 


3 


6 


205 


2 



T. 



216I 
216 



IQ9 




184 


rl 


ifc8 


u 


158 


b 


192 




163 


141 


rt 






132 


u 


rX 







Ph 


J 34 




112 




154 J 




35^ 


. 


47 


^ 


43 


c 


47 


a 


44 





41 


u 




(1) 


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x: 


29 


\t 


28 


•0 


43 




31 


c 


27 


d 




c 











23 


&i 



52 
181 

169 

51 



Table I. contains the results of the tests made by General 
Q. A. Gillmore at the Centennial Exposition, Philadelphia, in 
1876 (Van Nostrand's Magazine, March, 1877). The speci- 
mens were prepared " by mixing them dry in each case with 
an equal measure of clean sand, tempering the mixture with 
water to the consistency of stiff mason's mortar, and then 
moulding it into briquettes of suitable form for obtaining the 
tensile strength in a sectional area i^ inches square, equal to 
2]^ square inches. The briquettes were left in the air one 
day to set, then immersed in water for six days, and tested 
when seven days old. After thus obtaining the tensile strength 



Art, 37.] 



PHI LB RICK ' 6- EXPERIMENTS. 



355 



in each case, the ends of the broken specimens were ground 
down to i^ inch cubes, which were used the same day for 
obtaining the compressive strength by crushing." The col- 
umns "" No. tests " give the number of experiments from which 
were obtained the mean values contained in the columns C 
and T, 

C = Ult. Compressive resistance in lbs. per sq. in. 
T = '' Tensile '' " " " " " 

The former is given here in order to avoid the repetition of 
the makers' names hereafter. 

TABLE II. 





N. R. 


N. N. J. 


N. N. J. 


N. Y. n. 


L. H. 


H. C. 


AGE. 
















T. 


T. 


T. 


T. 


T. 


T. 


16 hours 


47 












24 hours 


55 


63 


58 


45 


44 


60 


36 hours 


55 












48 hours 


60 


64 




79 


50 


49 


60 hours 


68 












72 hours 













77 


4 days 










73 




7 days 


97 




71 


85 






10 days 








98 


105 




14 days 


120 












15 days 




134 




88 






19 days 


140 












21 days 











97 




25 days 












117 


I month 






92 




108 




64 days 








297 







69 days 








325 






2 months 






157 




202 




3 months 












175 


3i months. . . . 










221 




4 months 












177 


6 months 










250 




I year 











381 




l\ year 






241 








2 years 


257 




385 


336 


266 


292 



All specimens of neat cement mixed with fresh water at about 60° Fahr. 



356 



CEMENT AND BRICK IN TENSION, [Art. 3/. 



Table II. contains the results of a large number of tests of 
Rosendale cement (Trans. Am. Soc. of Civ. Engrs., Vol. VII., 
Feb. 1878 — " Improvement of the South Boston Flats," by 
Edward S. Philbrick), also that from Howe's Cave. 

" N. R." signifies '' Newark & Rosendale Cement Co." 
'' N. N. J." signifies '' Newark, N. J., Lime & Cement Co." 
'' N. Y. R." signifies " New York & Rosendale Lime & 
Cement Co." 

"L. H." signifies " Lawrence Cement Co. — Hoffman Rosen- 
dale Cement." 

*' H. C." signifies " Howe's Cave Cement." 

T = ultimate tensile resistance in pounds per square inch. 

The values of 7" are averages of from i to 2,217 tests. 
Table III. gives some results of the same neat cement 
specimens mixed with salt water at about 60° Fahr. 

TABLE III. 



AGE. 


N. N. J. 
T. 


L. H. 
T. 


AGE. 


N. N. J. 

T. 


L. H. 

T. 


24 hours .... 

2 days 

7 days 

10 days 

20 days 

I month . . . 


39 

56 

105 


19 

45 

50 

63 
79 


2 months, . 
3 J months. . 
6 months. . 

1 year .... 
i\ year .... 

2 years. . . . 


205 
311 


134 

275 
237 
367 

234 



The notation is the same as that of Table II. 



Art. 37.] 



MACLA Y'S EXPERIMENTS. 



357 



Experiments and Conclusions of Win. IV. Maclay, C. E. 

The following tables and conclusions are abstracted from 
** Notes and Experiments on the Use and Testing of Portland 
Cement," by Wm. W. Maclay, C. E. (Trans. Am. Soc. of Civ. 
Engrs., Vol. VI., Dec, 1877). 

He made many valuable experiments in order to determine 
the effect of different temperatures at different periods in the 
life of the cement. 

TABLE IV. 
Portland Cement. 



S w 


Q 
W 

!fi 

K 
W 
g 

td 
K 
14 






AVERAGE TENSILE STRENGTH 


PER SQ. IN. 




u. S 

a 

X u 

S 1 

el 


7 daj/s old. 

Temperature of cement paste when bri- 
quettes were moulded. 


21 days old. 

Temperature of cement paste when bri- 
quettes were moulded. 




32° 


40^ 


50° 


60° 


70° 


32° 


40° 


50° 


60° 


70° 


40" 




Lbs. 

156 
186 

259 
299 


Lbs*. 
147 
206 

275 
314 


Lbs. 
131 

183 

245 
299 


Lbs. 
133 
194 
240 
286 


Lbs. 
113 
143 
191 

254 


Lbs. 
265 
289 
348 
360 


Lbs. 

307 


Lbs. 
236 
292 
31S 
403 


Lbs. 
244 
260 

309 
386 


Lbs. 
212 

251 

282 

336 


60^ 




70^ 










40° 


40° 


45^ 


45° 


45° 


46^ 


46° 


43° 


43° 


43° 



Temperature of air during last 24 hours. 

Table IV. contains some of the results of Mr. Maclay's 
experiments which were made to determine the effect of the 
temperature of the water in which the specimens of neat 
cement were mixed. After the briquettes were moulded at 
temperatures shown in the upper horizontal column, and im- 
mersed in water either 7 or 21 days at the temperature shown 



35S CEMENT AND BRICK IN TENSION. [Art. 37. 

in the left vertical column, they were taken out and dried in 
air at the temperature shown in the lower horizontal row. 
The resistances in pounds are averages of five or more results. 

From these and many other similar results, Mr. Maclay 
concluded that the ultimate resistance follows " very closely 
the temperature of the water in which the sample briquettes 
were kept immersed, the warmest water giving the greatest 
tensile strength ; that a change from 40° to 70° in the water, 
increases the tensile strength of the briquettes of neat cement, 
seven days old, from 63 to 168 pounds per square inch, or from 
33 to 127 per cent. ; of the briquettes of mortar, gauged i to i, 
the same change in temperature increases them from 32 to 59 
pounds per square inch, or from 87 to 133 per cent.; of the 
briquettes gauged i cement to 2 of sand, from 19 to 37 pounds 
per square inch, or from 95 to 176 per cent. After an interval 
of three weeks the changes in tensile strength, . . . become 
less marked, and, in some cases, an increase in the temperature 
of the water diminishes the tensile strength." 

Other experiments seemed to " show quite conclusively 
that the tensile strength increases directly as the temperature 
of the air when the cement is being gauged, and inversely as 
the temperature of the air to which it is exposed for the last 
24 hours before breaking." 

" Exposing the briquettes after six days' immersion in 
water to a high drying temperature weakens them so invariably, 
that some interference with the setting seems clearly demon- 
strated " 

Some further experiments led him to conclude '' that Port- 
land cement gauged with either fresh or salt water, hardens 
more rapidly when immersed in salt or sewer water than in 
fresh water for the first seven days, and that this increase in 
the tensile strength probably continues for at least a year, 
more rapidly in the one than in the other.'' 

Table V. contains the results of Mr. Maclay' s tests on the 
relative influence of fine and coarse sand. 



Art. 37.] 



MA CLAY'S EXPERIMENTS. 



359- 



TABLE V. 

Portland Cement. 

Showing difference in mortars with fine and coarse sand. 



AGE OF 


1 VOL. CEMENT. 
•I VOL. SAND. 


1 VOL. CEMENT. 

2 VOLS. SAND, 


I VOL. CEMENT. 
3 VOLS. SAND, 




Fine sand. 


Coarse sand . 


Fine sand. 


Coarse sand. 


Fine sand. 


Coarse sand. 


I week .... 
I month . . . 


85 
162 


95 
202 


33 

81 


63 
94 


19 
49 


41 

74 



TABLE VL 
Portland Cement. 



w. 


BRAND. 


T. 


P.c. 


IOI.5 


Alsen & Son, Itzehoe, Germany. 


326 


93 


108.0 


<( t ( (< (i 


340 


91 


112 


Burham. 


289 


87 


113. 


( t 


317 


87 


114. 


(< 


285 


88 


115. 


Gibbs. 


2S0 


90 


116. 


Burham. 


316 


88 


117. 


t i 


301 


85 


iiS.o 


t( 


276 


85 


119.0 


< ( 


305 


85 


120.0 


( ( 


252 


84 


121. 


Saylor's American Portland. 


269 


90 


122.0 


( ( <( 1 ( 


281 


90 


123.0 




272 


90 


124.0 


( ( t.i (1 


260 


90 


126.0 




265 


90 


128.0 


< 1 ( t < ( 


369 


87 


152.0 




322 


78 



W = weight in lbs, per bushel. 

T = ultimate tensile resistance in lbs, per sq. in. 

P, c. = percentage that passed through sieve of 2,500 meshes per sq. in. 



360 



CEMENT AND BRICK IN TENSION. [Art. 37. 



** The deduction from this table is, that by increasing the 
fineness of the sand of which the mortar is made the tensile 
strength is diminished, and that this reduction in tensile 
strength increases with the amount of sand used in the mortar." 
An English experimenter, Lieut. W. Innes, R. E., was led to 
the same conclusion. 

Table VI. shows the results of experiments by. Mr. Maclay, 
made to determine the connection between the weight per 
bushel and tensile resistance of specimens seven days old. 
Commenting on the results he says, *' The close connection be- 
tween the weight . . . and the tensile strength . . 
is now proved to be very uncertain, if not entirely fallacious.'* 
As will hereafter be seen, these experiments invalidate a con- 
trary conclusion reached by the English experimenter, John 
Grant, M. Inst. C. E., in 1864. 

TABLE VII. 





< 

X 
K 

D 

CO 


FRANCIS. 


> 
III 


Port- 
land. 


GILLING- 
HA.M. 


PORTLAND. 


J. B. WHITE 
BROS. 


FRANCIS 
BROS. 


savlOr's 


AGE. 


Port- 
land. 


Port- 
land. 


Port- 
land. 


Neat 
cement. 


1 cement 
I sand. 


Roman. 


Medina. 


American 
Portland. 


1 week. 

2 " 

3 '' 

1 mo.. . 

2 '^ ... 

M::. 

i-:;: 

I " ;:; 
9;' ... 

10 ... 

1 yr.. 
li - .. 

2 "'' .. 

3 " •■ 

5 

6 " .. 

7 " • 


278 

256 
359 
332 
5^4 
525 
513 
554 
355 

304 

443 
611 
326 


184 
175 
302 

374 
4x6 

423 
432 
459 
327 

310 

414 
497 
355 


212 

182 
306 
30 r 
378 

383 
428 
426 
264 

270 

367 

339 
224 


250 

248 
380 

400 

431 

497 
529 

5i2 

312 

4^4 

569 
559 
476 


363 

416 
469 

523 

542 

546 

589 
584 
583 
580 
580 
59=" 


157 

201 
243 

284 

308 
318 

351 
349 
364 
364 
364 
384 


50 

77 

83 

116 

143 
210 

209 
286 

243 
268 
281 
279 
296 
315 


94 
135 

132 
136 

2ro 
183 

203 

212 

123 
122 
128 
136 
162 
168 


364 

413 
498 

525 

576 
575 

586 
599 



Vertical columns give tensile resistance in pounds per square inch. 



. Art. 37.] 



KEENE AND PARIAN CEMENTS. 



361 



Table VI I. shows the variation of ultimate tensile resist- 
ance, per square inch, of various neat cements (with one ex- 
ception) with age. The left vertical column shows the time 
during which the specimens were kept under water, and the 
other vertical columns the ultimate tensile resistance in pounds 
per square inch. 

The results for the Burham, Francis, Tingey, Gillingham 
and Saylor Portland cements are from Mr. Maclay's paper ; 
the others are from" Experiments on the Strength of Cements," 
London, 1875, by John Grant, M. Inst. C. E. ; all are means 
of great numbers of experiments. 

The mean results of Mr. John Grant's experiments on 
Keene's and Parian cements are given in Table VIII. 

TABLE VIII. 





keene's CEMENT. 


PARIAN CEMENT. 


AGE AND TIME IM- 
MERSED IN WATER. 


In water. 


Out of water. 


In water. 


Out of water. 




T. 


T. 


T. 


T. 


1 week 

2 " 


242 
216 
224 
218 
202 
226 


243 
260 
258 
260 
288 
320 


264 
267 
242 
242 
222 
232 


285 
298 
310 
332 
322 
380 


3 " 


1 month 

2 " 

3 " 



T = ultimate tensile resistance in pounds per square inch. 

As the result of his experiments on Portland and Roman 
cements Mr. Grant was led to the following conclusions : 



362 CEMENT AND BRICK IN TENSION. [Art. 37. 

1. Portland cement, if it be preserved from moisture, does not, like Roman 
cement, lose its strength by being kept in casks, or sacks, but rather improves by 
age ; a great advantage in the case of cement which has to be exported. 

2. The longer it is in setting, the more its strength increases. 

3. Cement mixed with an equal quantity of sand is at the end of a year ap- 
proximately three-fourths of the strength of neat cement. 

4. Mixed with two parts of sand, it is half the strength of neat cement. 

5. With three parts of sand, the strength is a third of neat cement. 

6. With four parts of sand, the strength is a fourth of neat cement. 

7. With five parts of sand, the strength is about a sixth of neat cement. 

8. The cleaner and sharper the sand, the greater the strength. 

9. Very strong Portland cement is heavy, of a blue-gray color, and sets slowly. 
Quick setting cement has, generally, too large a proportion of clay in its composi- 
tion, is brownish in color, and turns out weak, if not useless. 

TO. The stiffer the cement is gauged, that is, the less the amount of water used 
in working it up, the better. 

11. It is of the greatest importance that the bricks, or stone, with which Port- 
land cement is used, should be thoroughly soaked with water. If under water, in 
a quiescent state, the cement will be stronger than out of water. 

12. Blocks of brickwork, or concrete, made with Portland cement, if kept 
under water till required for use, would be much stronger than if kept dry. 

13. Salt water is as good for mixing Portland cement as fresh water. 

14. Bricks made with neat Portland cement are as strong at from six to nine 
months as the best quality of Staffordshire blue brick, or similar blocks of Bramley 
Fall stone, or Yorkshire landings. 

15. Bricks made of four parts or five parts of sand to one part of Portland 
cement will bear a pressure equal to the best picked stocks. 

16. Wherever concrete is used under water, care must be taken that the water 
is still. Otherwise, a current, whether natural or caused by pumping, will carry 
away the cement, and leave only the clean ballast. 

17. Roman cement, though about two-thirds the cost of Portland, is only about 
one-third its strength, and is therefore double the cost, measured by strength. 

18. Roman cement is very ill adapted for being mixed with sand. 

Mr. Don J. Whittemore has proposed the following formula 
for the ultimate tensile resistance of cements : 

in which T is the ultimate tensile resistance in pounds per 
square inch ; A, an empirical coefficient, and N the age of the 



Art. 37.] 



ARTIFICIAL STONES. 



363 





Fig.l 



-IVi- 



cement in days. For Portland cement (up to two years old) 

he gives x — 10, and y^ = 

267 to 356, by the aid of 

Mr. Grant's experiments. 

(See Trans. Amer. Soc. of 

Civ. Engrs"., Vol. VII., Sept. 

1878). 

Fig. I shows the bri- 
quette used by Mr. Maclay ; 
Fig. 2, that used by Mr. 
Grant, while that shown in 
Fig. 3 is the one generally 
used at the present time. 
Each briquette is i^ inches 
thick, giving a breaking sec- 
tion of i^ X i^ = 2.25 
square inches. In such test- 



is" 

y__ 



Fig.2 




ing it is very necessary that the pull should be central. 



Artificial Stones. 

The tensile resistances of many artificial stones and some 
natural British ones, can be found in '' A Practical Treatise on 
Natural and Artificial Concrete," by Henry Reid, London, 

1879. 

On page 198 he gives the following results of Professor 
Ansted's experiments, T being the ultimate resistance per 
square inch : 

Ransome stone (artificial) Z" = 360 pounds. 

Portland stone T ~. 201 " 

Bath stone T =■ 145 " 

Caen stone Z" = 140 " 

He also gives for '' Victoria " (artificial) stone, three months 
old, 



3^4 CEMENT AND BRICK IN TENSION. [Art. 3/. 

T = 740 pounds per square inch. 

From 35 experiments on *' rock concrete " pipe two years 
old, for drainage and sewage purposes, Mr. Reid found : 

HIGHEST. MEAN. ' LOWESY. 

T = 700 444 213 pounds per square inch. 

Bricks. 

Mr. Francis CoUingwood, C. E. (Trans. Amer. Soc. of Civ. 
Engrs., Vol. VII., Sept., 1878), found, as a result of twelve 
experiments on *' good Haverstraw stock brick," the following 
values : 



T = 358 169 90 pounds per square inch. 

Adhesion between Bricks and Cement Mortar. 

General Q. A. Gillmore ('* On Limes, Hydraulic Cements 
and Mortars ") cemented Croton bricks together crosswise and 
then separated them by a pull. He used pure cement paste 
and mortars of various proportions, by volume, of cement to 
sand, but never more sand than i volume of cement to 2 vol- 
umes of sand. Nearly all the cement was Rosendale, although 
some specimens were prepared with Hancock (Maryland) or 
James River cement. Bricks so cemented in pairs were kept 
320 days and then separated. Reviewing the results. Gen. 
Gillmore says, " In tearing the bricks apart, at the expiration 
of the time specified, in a majority of cases the surface of con- 
tact of the brick and mortar remained intact, the adhesion to 
the brick overcoming the cohesive strength either of the bricks 
themselves, or of the mortar composing the joint between 
them. The results, therefore, although interesting for other 
reasons, furnish no entirely satisfactory measure of the power 
of adhesion." 



Art. 38.] 



ADHESION TO BRICKS. 



365 



Also, *' At the age of 320 days (and perhaps considerably 
within that period) the cohesive strength of pure cement mor- 
tar exceeds that of Croton front bricks. The converse is true 
when the mortar contains fifty per cent., or more, of sand." 

TABLE IX. 



MORTAR OR PASTE. 


MATERIALS CEMENTED. 


ADHESION PER 
SQ. INCH IN LBS. 


RATIO OF ADHESION 

TO RESISTANCE OF 

PURE CEMENT. 


Pure cement 

I vol. cement i vol. sand. . 

I " " 2 " " .. 

I 4 

_ ( < It C ' ' ' ' 
^ ( ( ( ( /: 1 ( < < 

Pure cement 


Croton bricks. 
« t < ( 

<( (( 

< < ( ( 

( ( It 

Fine cut granite. 
( < ( ( 1 ( 

( ( 1 ( < t 

(( t ( it 

(< (I > 1 

1 


30 
15 

12 

6 

5 

4 

3 

27 

20 

12 

9 

7 


S 
7 
3 
8 
2 
3 
3 
5 
8 
6 
2 

9 


I.OO 

0.51 

0.40 
0.22 

0.17 
0.14 

O.II 
I.OO 


I vol. cement I vol. sand. . 
I " " 2 " " 

- C ( ( < - < ( ( ( 

. < 1 < ( 1 ' < < ' 
i 4 


0. 76 
0.46 

0-33 
0.29 



Table IX. contains the results of another series of experi- 
ments by General Gillmore, made for the purpose of determin- 
ing the adhesion to Croton front bricks and fine cut granite, 
of mortars containing different proportions of sand. " The 
bricks were used wet, and were well pressed together by hand. 
They were wetted with fresh water every alternate day for 29 
days, the age of the mortar when tested. Each result is the 
average of five trials." 



Art. 38.— Timber. 

Table I. contains the results of experiments made by Che- 
vandier and Wertheim (" Memoire Sur les Propri^tes M^canique 
du Bois ; " by E. Chevandier and G. Wertheim, 1846). The 



366 



TIMBER IN TENSION, 



[Art. 38. 



TABLE I. 



KIND OF WOOD. 



Hornbeam 
Aspen .... 

Alder 

Sycamore. 
Maple . . . 

Oak 

Birch 

Beech . . . . 

Ash 

Elm 

Poplar . . . 

Acacia . . . 

Fir 

Pine 



COEFFICIENTS OF ELAST. 



ELASTIC LIMIT. 



Pounds. 
1,335,000 
1,329,000 
1,021,000 
1,616,000 
1,459,000 
( 1,765,000) 

) '" \ 

( 1,214,000 ) 

( 2,431,000) 

\ to \ 

( 1,263,000) 

1,450,000 

( 1,798,000 1 

( 1,364,000) 

1,436,000 

( 1,027,000 ) 

] '° I 

( 901,000 1 
( 2,206,000 ) 
\ to - 

( 2,018,000 ) 

! 2, 2 18, 000 ) 
to ^ 

1,310,000 ) 

i,oSS,ooo 



Pounds. 
3,060 
4.380 
2,570 
3.270 
3.870 

3,340 



2,300 
3,300 
2,890 
2,620 
2,100 

4,540 

3,060 
2,320 



ULTIMATE TENSILE RESIST. 



Pounds. 
4,250 
10,240 (l) 
6,460 (I) 
8,760 (I) 
5,090 (l) 

8,530 



6,110 
5,080 
9,640 (l) 
9,940 (I) 
2,800 (l) 

11,280 

5,940 

3.530 



results are means, and were obtained from small, well seasoned 
rods, with cross-sectional areas varying from 0.30 square inch 
(some fir .specimens) to 1.50 square inches (one oak specimen), 
and are given in pounds per square inch. The results indi- 
cated thus "(i)," belong to one tree only, others, to several. 

The limit of elasticity is that force per square inch which 
will produce a permanent elongation of 0.00005 of the original 
length. 

These experimenters found that the elongations produced 
by different weights were composed of two parts, one perma- 
nent and one elastic ; the latter being essentially proportional 



Art. 38.] 



LA SLE TTS EXFERIMEN TS. 



367 



TABLE II. 



KIND OF TIMBER. 



Oak, English 

Oak, English ... 

Oak, French 

Oak, Dantzic 

Oak, American White .... 
Oak, American, Baltimore 
Oak, African (or Teak) . . . 
Teak, Moulmein ........ 

Iron Wood, Burmah 

Chow, Borneo 

Greenheart, Guiana 

Sabicu, Cuba 

Mahogany, Spanish 

Mahogany, Honduras 

Mahogany, Mexican. . . 
Eucalyptus, Australia : 

Tevvart 

Mahogany 

Iron-Bark 

Blue Gum 

Ash, English 

Ash, Canadian 

Beech 

Elm, English 

Rock Elm. Canada 

Hornbeam. England 

Fir, Dantzic 

Fir, Riga 

Fir. spruce, Canada 

Larch, Russia 

Cedar, Cuba 

Red pine, Canada 

Yellow pine, Canada ... . 

Yellow pine, Canada 

Pitch pine, American. .. . 
Kauri pine. New Zealand. 
Georgia pine, American.. 

Locust, American 

White oak, American 

Spruce, American 

White pine, American. . . . 
Hemlock 



EXPERIMENTER. 



Laslett. 



Hatfield 



0.858 
0.893 
0.976 
0.838 
0.969 
0.742 
0.971 
0.777 
1 . 176 

1-134 
1. 141 

0.917 
0.765 
0.659 
0-655 

1. 169 
0.996 
1. 150 
I 049 
0.750 
0.588 
0.705 
0.642 
0.748 
0.819 
0.603 

0.553 
0.484 
0.649 
0.469 
0.553 
o 551 
0.552 
0.659 
o 544 



ULT. RESIST. IN 

POUNDS PER SQ. 

INCH. 



3,837 
7.571 
8,102 
4.217 
7,021 
3.832 
7.052 
3.501 
9,656 
7,199 

8,8co 

5,558 

3-791 
2,998 

3.427 



2,940 

8-377 

6,048 

3,780 

5,495 

4.853 

5,460 

9,182 

6,405 

3.231 

4.051 

3-934 

4,203 

2,870 

2,705 

2.759 

2.259 

4,666 

4,040 

16,000 

24,800 

'9,500 

19,500 

12,000 

8,700 



B 

IN POUNDS PER 
SQ. INCH. 



3 

a, 

a 

o 
u 

•c 
c 

3 
o 



to the load and the former measurable even for small loads 
and variable not only with the load but also with the time 
during which the load acted : that the coefficient and limit of 
elasticity augmented with the seasoning, but that the greatest 
elongation diminished under the same circumstances ; that if 
the coefficient of elasticity and ultimate resistance along the 



368 



TIMBER IN TENSION. 



[Art. 38. 



fibre be taken as units, the coefficients of elasticity along the 
radius and tangent to the tree, will be 0.165 and 0.091 respect- 
ively, while the ultimate resistances in the same directions 
will be respectively o. 163 and 0.159, these results being con- 
sidered averages. 

The ultimate tensile resistances of many woods, domestic 
and foreign, are given in Table II., as well as the specific grav- 
ities. 

The column '' B'' will be explained hereafter, in the chap- 
ter on transverse resistance or bending. 

TABLE III. 





ULT. RESIST. 


ELASTIC LIMIT 


COEFFICIENT OF 


PER CENT. 


OF EXTEN- 


WOOD. 


IN POUNDS PER 


IN POUNDS PER 


ELASTICITY IN 


SION 


AT 




SQ. IN. 


SQ. IN. 


POUNDS PER 


















SQ. IN. 


Elas. Limit. 


Fracture. 


White Pine 


6,880 


3,900 


183,490 


0.40 


0.73 


Yellow Pine .... 


20,700 


13,200 


240, 240 


0.63 


1.65 


Locust 


28,930 
9,790 


19,200 
5,700 


373. S30 
213,520 


1 . 10 


1.85 
0.85 


Black Walnut . . 


0.53 


White Ash 


15,490 


9,700 


206,540 


0.78 


1.48 


White Oak 


13,210 


8,100 


220,130 


0.77 


1.30 


Live Oak 


10,310 


6,300 


247,510 


0.58 


I-I5 



The table gives average results. Those determined from 
experiments of Mr. Laslett are of English origin ('' Timber 
and Timber Trees, Native and Foreign,'' by Thomas Laslett, 
1875); the others are from American experiments by the late 
R. G. Hatfield ('' Transverse Strains," 1877). Mr. Laslett's 
specimens were 2 inches square in cross section, and genearlly 



Art. 38.] TESTS BY THURSTON AND LAIDLEY. 369 

were 30 inches long, while those of Mr. Hatfield were about 
0.35 inch round. 

It will be observed that Mr. Hatfield reached far higher 
results than Mr. Laslett. This disagreement may be due to 
the larger cross-sectional area of the latter's specimens, which 
certainly brings his (Mr. Laslett's) results more nearly in 
accordance with what might be expected from such pieces as 
are ordinarily used by engineers. Mr. Hatfield's specimens 
were far too small for technical purposes. 

Table HI. is taken from a paper *^ On the Strength of 
American Timber," by Prof. R. H. Thurston (Jour. Frank. 
Inst., Oct., 1879). ^^^ specimens were turned down to about 
0.5 inch diameter for a length of 4.00 inches. 

The small values of the coefificient of elasticity, as compared 
with those given in Table I., are probably due to the fact that 
they were found at the elastic limit. Smaller intensities of 
stress would probably give much larger values. 

Prof. Thurston also states that timber in tension takes a 
permanent set however small the intensity of stress. 

The values given in Table IV. were found by Col. Laidley, 
U. S. Army, in the Government machine at Watertown, Mass. 
(Ex. Doc. No. 12; 47th Congress, 2d Session). Two of the 
specimens were about 0.63 inch in diameter, and one 1.25 
inches. All the rest possessed diameters of about one inch 
each. 

Such small specimens as those of Hatfield, Thurston, and 
Laidley, which were probably selected, give much larger results 
than would be found for large pieces of ordinary lumber ; these 
considerations are highly prejudicial to the technical value of 
the results. 

Far more Importance attaches to the matter of size and 
character of timber specimens than to those of metallic ones. 
In the latter there is at least an approach to homogeneity of 
material, which the presence of knots, conditions of growth, 
seasoning, and other influences effectually prevent in timber 
24 



370 



TIMBER IN TENSION. 



[Art. 38 



specimens. Hence it is the more necessary to test timber in 
circumstances of condition and size as nearly identical as pos- 
sible with those which attend its actual use. 



TABLE IV. 
Diameter of Test Specimens, i inch. 



NO. 


KIND OF WOOD. 


ULTIMATE RSISTANCE PER SQL'ARE INCH IN 
POUNDS. 


NO. OF 




Greatest. 


Mean. 


Least. 




I 


Yellow Pine 


17,922 


15,478 
13,810 
16,160 

8,916 
14,283 

6,787 
14,313 
11,164 

11,492 
18,682 
24,120 
11,632 
17,410 
10,124 
20,390 
15,995 


12,066 


4 
I 
I 
4 
4 
2 

4 
3 
3 
3 

3 
3 
4 
3 
2 


2 

3 
4 
5 
6 

7 
8 

9 
10 


Oregon Pine 

Oregon Spruce 

White Pine 




11,299 

17,044 

7,466 

19,400 

15,71-1 
14,650 

22,838 

27,532 

11,733 
22,703 

12,133 
20,520 
19,610 


5.300 

11,600 

6,107 

4,586 

7,312 

9,286 

13,885 

18,961 

10,667 

12,670 

7,600 

20,260 

12,400 


Spruce 


White Wood 

Gum Wood 


W^hite Maple 

Black Walnut 

Red Birch 


II 


White Ash 


12 


Brown Ash 


13 
14 
15 
16 


White Oak 


Red Oak 


Yellow Oak 


Hickory 


3 







All specimens were of well seasoned wood. 



CHAPTER VI. 

Compression. 

Art. 39. — Preliminary. 

With the exception of material in the shape of long col- 
umns, but few experiments, comparatively speaking, have been 
made upon the compressive resistance of constructive materials. 

Pieces of material subjected to compression are divided 
into two general classes — " short blocks " and " long columns ; " 
the first of these, only, afford phenomena q{ pure compression. 

A "• short block " is such a piece of material, that if it be 
subjected to compressive load it will fail by pure compression. 

On the other hand, a long column (as has been indicated in 
Art. 25) fails by combined compression and bending. 

Short blocks, only, will be considered in the articles imme- 
diately succeeding, while long columns will be separately con- 
sidered further on. 

The length of a short block is usually about three times its 
least lateral dimension. 

It has already been shown in Art. 4 that the greatest 
shear in a short block subjected to compression, will be found 
in planes making an angle of 45° with the surfaces of the 
block on which the compressive force acts, i, e., with its ends. 
If the material is not ductile, this shear will frequently cause 
wedge-shaped portions to separate from the block. But the 
friction at these end surfaces, and in the surfaces of failure 
will prevent those wedge portions shearing off at that angle. 
In fact the friction will cause the angle of separation to be 



3/2 WROUGHT IRON IN COMPRESSION. [Art. 40. 

considerably larger than 45° ; let it be called a. Then, in 
order that there may be perfect freedom in failure, the length 
of the block must not be less than its least width or breadth 
multiplied by 2 tan a. In some cases, a has been found to be 
about 55°, for which value 

2 tan a = 2 X I.43 = 2.86. 

It was shown in the first section of Art. 32, that the 
*' ultimate resistance " to tension is in reality a mean, and not 
the greatest intensity which the material exerts. The same 
course of reasoning will show that it is, also, in general, im- 
possible to subject a short block to a uniform intensity of 
compression throughout its mass, and that the " ultimate re- 
sistance to compression " is a mean, usually considerably less 
than the greatest intensity which exists at the centre of a 
normal section. As the inner portion will be supported later- 
ally by that outside of it, large blocks of brittle material may 
give greater intensities of ultimate resistance than small ones. 



Art. 40. — Wrought Iron. 

It is difficult to fix the point of failure of a short block of 
wrought iron or other ductile material. An excessive compres- 
sive force causes the material to increase very considerably 
in lateral dimensions, or to " bulge " out, so that every in- 
crease of compressive force simply produces an increased area 
of resistance, while the material never truly fails by crumbling 
or shearing ofT in wedges. 

A short block of wrought iron is usually considered to fail 
when its length is shortened by five to ten per cent. 

If /j is any intensity of stress while /^ is the compressive 
strain, or shortening per unit of length caused by p^, then 
according to Eq. (2) of Art. 2, the coefficient of elasticity for 
compression at the intensity /j, will be 



Art. 40.] 



COFFICIENTS OF ELASTICITY. 



373 



77 _ /i 



(■) 



This ratio is not constant for all degrees of stress and strain, 
though for wrought iron, within the elastic limit, the diver- 
gences from a mean value are not great. Table I. contains 
coefficients of elasticity calculated by Prof. De Volson Wood, 
in the manner shown by Eq. (i), from the data determined by 
Mr. Eaton Hodgkinson and given in his work before cited. 
(See Prof. Wood's " Treatise on the Resistance of Materials "). 

TABLE I. 



Ap,. 


LI,. 


■^1- 


LI,. 


E,. 


Pounds. 


Inch. 


Pounds. 


Inch. 


Pounds. 


5,098 


0.028 


20,796,500 


0.027 


21,864,000 


9.578 


052 


21,049,000 


0.047 


23,595,000 


14,058 


0.073 


21.979,000 


0.067 


24,273,000 


16,298 


0.085 


21,343.000 










18,538 


0.096 


22,156,000 


0.089 


24,108,000 


20,778 


0.107 


22,160,000 


O.IOO 


24,038,000 


23,018 


0.II9 


23.587,000 


0. 113 


23,587,000 


25,258 


0.130 


22,095,000 


0.128 


23,679,000 


27,498 


0.142 


22.111,000 


0.143 


22,259,000 


29.738 


0. 152 


21,938,000 


0. 163 


21,139,000 


31,978 


0.174 


20,979,000 


0. 190 


19,478,000 



The results belong to two square bars, and E^ is in pounds 
per square inch. A is the area of cross section ; it was 1.0506 
square inches for the first bar and 1.0363 square inches for 
the other. Hence the bars were about one inch square. They 
were also ten feet long (Z — 10.00 feet) and required lateral 
support to be kept in alignment so as to act like short 
blocks. 

The table shows that the values of E^ increase with/,, when 
the latter is small ; an opposite result was found for tension. 

What may be called the elastic limit is found for p, = 



374 



WROUGHT IRON IN COMPRESSION. [Art. 40. 



30,000.00 pounds per square inch (nearly). Hence, it is seen 
that the greatest value of E^ is found for p^ equal to one-half 
to two-thirds the elastic limit. 

The same general remarks in regard to the elastic limit, 
which were made in connection with tension, may be also ap- 
plied to the compressive elastic limit. 

The " Steel Committee " of British civil engineers, in 1870, 
made some experiments on twelve bars of Lowmoor wrought 

TABLE II. 



POUNDS PER 


SQUARE INCH FOR 


Elastic Limit. 


Coefficient of Elasticity. 


Pounds. 


Pounds. 


29,800 


29,091,000 


25,800 


29,091,000 


29,100 


28,718,000 


26,200 


28,000,000 







iron, 1.5 inches in diameter and 120 inches long. These twelve 
experiments were divided into four sets of three each, and the 
table gives the means of each of these sets or groups. The co- 
efficients are computed at the elastic limit. Judging from the 
results in Table I., smaller values of p^ would have given 
larger values of E^, 

As a mean value, the coefficient of elasticity for wrought 
iron in compression may be taken at 28,000,000 pounds per 
square inch. For every ton (2,000.00 pounds) of compression 
per square inch, therefore, a piece of wrought iron will be 
shortened by an amount equal to 



Art. 40.] 



ULTIMATE RESISTANCE. 



375 



2,000 



28,000,000 14,000 



of its length. 



Table III. contains the results of some experiments made 
by Mr. Kirkaldy on some specimens of Swedish iron, in 1866. 
The last column gives the per cent, of compression of original 
length which the piece suffered at the point called the " ulti- 
mate compressive resistance." The results show well the great 
increase of resistance which a short block of ductile material 
offers with the increase of compression. 

TABLE III. 



SECTION OF 


LKNGTH. 


POUNDS PER SQUARE INCH FOR 


PFR CENT. 


SPECIMEN. 


Elas. Lim. 


Ult. Resist. 


COMPRESSION. 


In. 


Ins. 


Lbs. 


Lbs. 




15 


1.5 


24,050 


148,800 


45 


15 


1-5 


21,200 


28,100 


4 


15 


3 


23,300 


84,900 


33 


1.0 □ 


I 




184,100 


53 


» 



Table IV. gives the results of experiments on some very 
short lengths of Phoenix and Keystone columns. The first six 
results are for Phoenix sections from experiments by the Phoe- 
nix Iron Co., in 1873 ; the two following are for the same sec- 
tion from experiments made at Watertown, Mass., in 1879; 
while the last result belongs to a Keystone section experi- 
mented upon by Mr. G. Bouscaren, in 1875. Unfortunately 
the amount of compression or shortening, in each instance, 
was not recorded. 

Reviewing the results given in Tables II., III. and IV., it is 
seen that the "elastic limit " of wrought iron in compression, in 



37^ 



CAST IRON IN COMPRESSION. 



[Art. 41. 



TABLE IV. 





RATIO OF LENGTH TO 


AREA OF SECTION, 


ULT. RESIST. IN POUNDS 


LENGTH, INCHES. 










DIAMETER OF SECTION. 


SQ. IN. 


PER SQ. IN. 


8.00 


1.46 


6.97 


60,570.00 


8.00 


1.46 


6.97 


60,390.00 


4.00 


0.92 


5.62 


65,870.00 


4.00 


0.92 


5.62 


65,870.00 


4.00 


I. 01 


2.92 


56,890.00 


4.00 


I. 01 


2.92 


55,560.00 


8.00 


I. 00 


ir.90 


57,130.00 


8.00 


I. 00 


11.90 


57,300.00 


9.00 


I. 12 


14-25 


51,500.00 



short blocks, may be taken from 0.4 to 0.5 its ultimate com- 
pressive resistance, while the latter may be taken at about 
60,000.00 pounds per square inch. 



Art. 41. — Cast Iron. 

The irregular elastic behavior of cast iron, as seen in ten- 
sion, will also be discovered in compression. Table I. contains 
results computed from the data obtained by Captain Rodman 
by testing solid cylinders 10 inches long and 1.382 inches in 
diameter. The second column belongs to a specimen cylinder 
taken from a lo-inch columbiad, and the third or last to a trial 
cylinder of remelted Greenwood and Salisbury iron. Neither 
specimen can be considered to possess a true elastic limit, but 
what is ordinarily so termed may be taken at about 20,000 
pounds per square inch. 

In the first specimen the first permanent set took place 
at 3,000.00, and in the second at 5,000.00 pounds per square 
inch. 



Art. 41.] 



COEFFICIENTS OF ELASTICITY. 



Z77 



TABLE I. 



INTENSITY OF STRESS. 


COEFFICIENT OF ELASTICITY IN POUNDS PER SQUARE INCH. 


1,000 


6,896,600 






2,000 


8,888,900 


33,333,300 


3,000 


9,836,100 


18,750,000 


4,000 


10,666,700 


13,793.100 


5,000 


10,752,700 


13,888,900 


6,000 


11,320,800 


12,766,000 


7,000 


11,382,100 


13,725,500 


8,000 


11,510,800 


13,559,300 


9,000 


11,920,500 


13,432,800 


10,000 


T2, 121, 200 


13,333,300 


11,000 


12,290,500 


13,095,200 


12,000 


12,182,700 


13,186,800 


14,000 


12,444,400 


13,207,500 


16,000 


12,260,500 


12,903.200 


18,000 


11,920,500 


12,857,100 


20,000 


11,695,900 


12,578,600 


22,000 


11,253,200 


12,290,500 


26,000 


10,236,200 


11.607,200 


30,000 


8,596,000 


10,101,000 


35,000 
40,000 




7,658,600 
5,333,300 





For a bar ten feet long and one inch square, Mr. Eaton 
Hodgkinson found the following values : 



GREATEST. 
13,216,000.00 



MEAN. 
,12,134,100.00. 



LEAST. 

10,837,100.00 ; 



all in pounds per square inch. The greatest value was found 
at 2,240, and the least at 38,080 pounds per square inch. 

Since the coefficient of elasticity measures the stiffness of a 
body, and since the coefficient of elasticity for wrought iron in 
compression has been seen to be at least twice as great as that 
of cast iron in the same condition, wrought iron is at least 
twice as stiff, compressively, as cast metal. A bar of the latter 
material will be compressed by 2,000 pounds per square inch, 
about 



37^ CAST IRON IN COMPRESSION. [Art. 41. 

2,000 I r ., 1 .1 

■ — = -p ■ of its leng^th. 

12,000,000 6,000 

If / is the length of a bar in inches, W the compressive 
stress in pounds per square inch, then Hodgkinson found the 
total decrement in inches for lo-feet Low Moor cast-iron bars 
to be 

V — /(0.012363359 — V0.000152853 — 0.00000000191212/^F). (i) 

and the permanent set, in inches : 

0.543^" + 0.0013 (2) 

Major Wade tested a number of specimens of cast iron of 
different numbers of fusions, in order to determine the ultimate 
compressive resistance. His specimens were from 0.5 to 0.6 
inch in diameter, and from 1.25 to 1.5 inches (nearly). The 
results were as follows : 

FUSION. NO. OF EXPS. GREATEST. MEAN. LEAST. 

2d 4 114,504 99^770 : 84,529 

3d 2 140,415 139.540 138,666 

2d and 3d... 2 169,427 168,589 167.752 

2d 2 140,415 136,868 133,321 

3d..- 1 168,251 168,251 168,251 

2d 5 163,528 154,576 144,141 

3d 4 t74jI20 167,030 156,863 

All results are in pounds per square inch. 

As the specimens gave way, portions sheared off along 
planes making angles with the normal sections of specimens 
varying from 46° to 62.5°. This is the characteristic compres- 
sive fracture of cast iron. 

The 3d fusion iron gave the highest resistance. 

Mr. Hodgkinson (" Report of the Commissioners appointed 
to Inquire into the Application of Iron to Railway Purposes," 
1849) took specimens of 16 different kinds of British irons, 0.75 



Art. 41.] 



UL TIM A TE RESISTANCE. 



379 



inch in diameter and 0.75 and 1.5 inches long, with the follow- 
ing results : 

GREATEST. MEAN. LEAST. 

117,605 86,284 56,445 pounds per square inch. 

As a rule, the short specimens gave from 5 to 10 per cent, 
greater resistance than the longer ones. From another set of 
experiments with 22 different kinds of iron (specimens 0.75 
inch in diameter and 1.5 inches long) he found : 

GREATEST. MEAN. LEAST. 

115,995 84,200 54j76i pounds per square inch. 

Mr. Hodgkinson found that the hardness and ultimate 
crushing resistance of thin castings were greatest near the 
surface, but that in thick castings the surface and heart gave 
essentially the same results. He also found that thin castings 
gave considerably greater ultimate resistance to crushing than 
thick ones. 

Sir Wm. Fairbairn tested the effect of remelting on " Eg- 
linton " No. 3, hot-blast iron with the following results : 



REMELTINGS. ULT. RESIST. 

1 98,560 

2 97,660 

3 92,060 

4 91,170 

5 92,060 

6 92,060 

7 91,620 

8 92,060 

9 123,420 



REMELTINGS. ULT. RESIST. 

10 129,250 

II 156,350 

12 163,740 

13 147,840 defec. 

14 214,820 

15 171,810 

16 157,920 

17 

18 197,190 



All results are in pounds per square inch. It is observed 
that the 14th remelting gives the highest resistance. 

From what precedes, it is seen that the ultimate compres- 
sive resistance of cast iron, in good ordinary castings, may 
safely be taken from 85,000 to 100,000 pounds per square inch. 



380 



STEEL IN COMPRESSION. 



[Art. 42. 



Art. 42. — Steel. 

Table A. contains results taken from Prof. Woodward's 
history of the St. Louis arch. The elastic limit and ultimate 
resistance have been given here, as well as the coefficient of 
elasticity, in order to avoid reproduction hereafter. The 
column ^'/-^ ^" gives the ratio of length over diameter ; the 
latter varied from about a half inch to 1.14 inches. It will be 
seen from this ratio that the specimens, in many cases, were 
somewhat longer than '' short blocks." Prof.- Woodward gives 
many other results with yet greater ratios of / -r <^, in some of 
which E^ reached 38,000,000. 

TABLE A. 



MAKERS. 



, Brooklyn, 
Brooklyn. 



Am. Tool Steel Co. 
Am. Tool Steel Co. 

Wm. Butcher 

Am. Tool Steel Co., Brooklyn 
Parks Bros., Pittsburg 



Wm. Butcher 

Wm. Butcher 

Butcher Steel Works. 



Wm. Butcher 
Wm. Butcher 



C'ld chisels. 
C'ld chisels. 

No. 6. 
Lathe No. 4 
Boiler plate. 



Ingot of 
Chrome 
Steel. 



Ingot of 
Chrome 
Steel. 

Chrome 
Steel plate. 



l + d. 



5 



POUNDS PER SQ. INCH, 
AT 



Elastic 
Limit. 



50,600 
50,000 
38,000 
53.000 
34,000 
40,000 
23,000 
27,700 
39,600 
43^520 
37 400 
38,900 
45,400 
39,700 
50,030 

50»030 
50,120 
50,120 



Ultimate 
Resist. 



145,000 

148,000 

96,200 

105,000 



134,000 
132,800 

136,490 

122,300 

92,000 

102,000 

106,000 

80,050 

84,050 

86.200 

78,180 

76,500 

92,470 



COEFFICIENT OF 

ELASTICITY, LBS. 

PER SQ. INCH. 



20,200,000 
31,000,000 



13,514,000 
8,506,000 
13,585,000 
15,450,000 
11,800,000 
15,370,000 
13,226,000 
14.634,000 
13,937,000 
14.371,000 
16,233,000 



In 1868, Chief Engineer Wm. H. Shock, U.S.N., tested 



Art. 42.] RESISTANCE AND ELASTICITY. 3^1 

Specimens 3,5 inches long and half an inch square, of Parker 
Bros. " Black Diamond " steel with the following results : 

Normal untempered steel: Ult. Resist, from 100,100 to 
112,400 pounds per square inch. 

Heated to light cherry-red and plunged in oil at 82° Fahrr. 
Ult, Resist, from 173,200 to 199,200 pounds per square inch. 

Heated as before, and plunged in water at 79° Fahr., with 
final temper (plum-blue) drawn on heated plate : Ult. Resist. 
from 325,400 to 340,800 pounds per square inch. 

Heated as before and plunged in water at 79° Fahr.j and 
tested at maximum hardness: Ult. Resist, from 275,640 to 
400,000 pounds per square inch. In each of these cases there 
were three tests. 

The following values (each is a mean of 8 tests) were found 
by the United States Test Board, '' Ex. Doc. 23, House of 
Rep., 46th Congress, second Session," for small annealed speci- 
mens of tool steel, of about one inch in length and 0.715 inch 
in diameter : 

Ult. comp. persq. in. ) 175,99- I 174.586 ; 183,938 ; 193,413 ; 193,197 ; 174,586 ; 
of original section. ) 193,517 ; 174,895 pounds per square inch. 

Final comp. per sq. ) 134,717 ; 127,579 ; I49'88i ; 139,196 ; I45,75i I 128,834 ; 
in. of final section. ) 125,126 ; 140,489 pounds per square inch. 

The final lengths varied from 56 to 89 per cent, of the 
originals. 

Kirkaldy's " Experimental Inquiry into the Mechanical 
Properties of Fagersta Steel," 1873, furnish data from which 
may be computed a series of values of the ratio [E^ of stress 
over strain, or coefficient of elasticity, for different intensities 
of stress. 

All the specimens were cut from plates of mild steel of the 
thickness shown in the table, and were 100 inches long and 
2.25 inches wide. They were laterally supported in a trough 
arrangement designed by Mr. Kirkaldy. 



382 



STEEL IN COMPRESSION. 



[Art. 42. 



TABLE I. 





COEFFICIENTS OF ELASTICITY IN POUNDS PER SQUARE INCH. 


INTENSITY 
OF STRESS. 


Unannealed. 


Annealed. 




X inch plate. 


%-inch plate. 


>i-inch plate. 


%-inch plate. 


10,000 
14,000 
18,000 
22,000 
26,000 
30,000 
34,000 
38,000 
42,000 
46,000 
50,000 


58,824,000 
56,000,000 
54,545,000 
48,889,000 
45,614,000 
42,254,000 
40,000,000 
38,384,000 
35,000,000 
30,872,000 
25,000,000 


38,462.000 
33,333,000 
31,034,000 
29,730,000 
28,261,000 
24,000,000 
3,795,000 


50,000,000 

48,276,000 

45,000,000 

38,596,000 

23,214,000 

7,792,000 

5,207,000 

4,265,000 

3,717,000 


34,483,000 
33,333.000 
31,034,000 
26,190,000 
19,118,000 
2,542,000 

























Below 18,000 pounds per square inch annealing does not 
much change the coefficients for the ^''^-inch specimens, but 
affects the thin ones more decidedly. In all the specimens the 
elastic behavior is very irregular, and none of them can be said 
to possess a true elastic limit. In the unannealed thin and 
thick plates, the first permanent sets took place at 40,000 and 
20,000 pounds per square inch, respectively ; in the correspond- 
ing annealed ones, at 30,000 and 20,000, respectively. 

By referring to Table III. of Art. 34, it will be seen that 
the coefficients for compression are considerably larger than 
those for tension. 

Table II. contains coefficients of elasticity computed from 
the results of experiments made under the supervision of the 
" Steel Committee of Civil Engineers " (English). All are 
computed for the "■ limit of elasticity." The upper portion of 
the table belongs to round'specimens 1.382 inches in diameter 
and 50 inches (j^d diameters) long, tested in 1868 ; the lower 



Art. 42.] 



LIMITS OF ELASTICITY. 



383 



TABLE 11. 




QUALITY. 


GREATEST ^j. 


MEAN jEj. 


LEAST E^. 


Hammered ^ 

and >■ 

rolled. ) 


Bessemer. 
Crucible. 


26,130,000 
34,461,000 


35,000,000 
33,939,000 


34,461,000 
31,464,000 


Chisel, ) 

tire, rod, >• 

rolled, etc. ) 


Bessemer. 
Crucible. 


30,270,000 

31,550,000 


29,091,000 
29,474,000 


28,718,000 
28,000,000 



Ex = coefficient of compressive elasticity in pounds per square inch. 

portion to 1.5 inch round specimens and 120 inches long, 
which were tested in 1870. Table I. shows that considerably 
different results might be expected with lower intensities of 
stress. 

TABLE III. 





LIMITS OF ELASTICITY IN POUNDS PER SQUARE INCH. 


LENGTH. 




Greatest. 


Mean. 


Least. 


Inches. 


Bessemer; tires, ham-" 
mered, rolled, fagot- 
ted, etc. \ 

1 
Diam. = 1.382 ins. J 


53,520 
52,240 
49,800 


50,250 
48,540 
46,700 


42,500 
40,990 
40.470 


1.38 

2.76 

5-53 


Crucible; hammered," 
rolled, chisel, tires, 
rods, etc. 

Diam. = 1.382 ins. 


60,260 
59,000 

53,740 


55,760 
53,780 
49,860 


48,990 

43,990 
42,000 


1.38 

2.76 

5-53 



The upper Bessemer results are for a set of 18, and the 



384 



STEEL IN COMPRESSION. 



[Art. 42. 



lower for a set of 1 1 tests ; the upper crucible results are for a 
set of II, and the lower for a set of 20 (?) tests. 

The " limits of elasticity " of specimens of the same steels 
to which the upper portion of Table II. belongs (and for the 
same number of experiments) are shown in Table III. 

The following '' limits of elasticity," in pounds per square 
inch, correspond to the lower portion of Table II. : 

GREATEST. MEAN. LEAST. 

Bessemer 47,490 39,370 35,840 pounds. 

Crucible 60,480 52,190.. 36,290 pounds. 

TABLE IV. 





ELASTIC LIMIT IN POUNDS PER SQUARE INCH. 




1,2. 


0.9. 


0.6. 


0.3. 


1 diam. 

2 " 

4 " 
8 " 


64,000 
63,330 
62,330 
61,670 


62,670 
58,670 
58,670 
58,000 


60,000 
57,330 
53,330 
52,670 


39,000 
42,000 
41,000 
40,670 


Means. 


62,833 


59.500 


55,830 


40,670 




ULTIMATE RESISTANCE IN POU 


NDS PER SQUARE INC 


H. 


1 diam. 

2 " 

4 '• 
8 " 










169,910 
133,330 
102,170 


173,290 

117,560 

95,210 


156,000 

105,330 

84,830 


121,330 
81,760 
47,510 


Means. 


135,140 


128,690 


115,387 


83.540 



The results of the experiments of Mr. Kirkaldy on speci- 
mens of different grades of Fagersta steel and of various 
lengths in terms of diameter, are given in Table IV. All the 



Art. 42.] 



UL TTMA TE RESISTANCE. 



385 



specimens were turned to 1.128 inches (i square inch area) in 
diameter, and were of the lengths shown. 

The numbers 1.2, 0.9, 0.6 and 0.3 were used to indicate the 
different grades of steel, the larger numbers belonging to the 
higher steels. 

The specimens of one diameter in length shortened, under 
a load of 200,000 pounds per square inch, 21, 22, 26 and 48 per 
cent., respectively, for the marks 1.2, 0.9, 0.6 and 0.3. Three 
of the " 2 diam." specimens failed by detrusion, or by portions 
shearing off obliquely ; all the others either bulged or took a 
skew form, though one of the *' 8 diams." finally broke. 

Table V. contains the results of Major Wade's experiments 

TABLE V. 



DESCRIPTION. 


LENGTH OVER DIAMETER. 


ULT. RESIST. IN POUNDS PER 
SQUARE INCH. 


Not hardened 


2 55 

2.47 
2.52 
2.48 


198,944 

354,544 

391,985 
372,598 


Hardened, low temper 

Hardened, mean temper . . . 
Hardened, high temper .... 



All specimens about i inch long and 0.4 inch in diameter. 



on specimens of cast steel, in 185 1. The results are seen to be 
very high. 

A piece of the Hay steel used by Gen. Smith in the Glas- 
gow, Mo., bridge, about 154^ inches square and 3}^ inches long, 
gave an ultimate compressive resistance of 139,350 pounds per 
square inch (" Annales des Pouts et Chauss^es," Feb., 1881). 



386 



COPPER, ETC., IN COMPRESSION. 



[Art. 43. 



Art. 43. — Copper, Tin, Zinc, Lead and Alloys. 

Table I. shows some coefficients of elasticity (i.e., ratios 
between stress and strain), computed from data determined by 
Prof. Thurston, and given by him in the " Trans. Amer. Soc. 
of Civ. Engrs.," Sept., 1881. The gun bronze contained cop- 
per, 89.97 ; tin, 10.00 ; flux, 0.03. The cast copper was cast 
very hot. 

TABLE L 





COEFFICIENTS OF ELASTICITY 


IN POUNDS PER SQUARE INCH. 


STRESS IN POUNDS PER SQUARE 






INCH. 








Gun Bronze. 


Cast Copper. 


1,620 




1,254,000 




3,260 


3,622,000 


1,415,000 


6,520 


4,075,000 


1,651,000 


9,780 


6,113,000 


1,795,000 


13,040 


6,520,000 


1,824,000 


16,300 


5,433,000 


1,842,000 


19,560 


5,148,000 


1,845,000 


22,820 


3,935,000 


1,735,000 


26,080 


2,308,000 


1,503,000 


29,340 




1,144,000 




32,600 


1,073,000 


815,000 


48,900 


463,600 


332,500 



The ratios of stress over strain are far from being constant. 
Strictly speaking, therefore, there is no elastic limit in either 
case. In that of the gun bronze, however, it may be approxi- 
mately taken at 20,000 pounds per square inch (Prof. Thurston 
takes it 22,820), and in that of the copper at 25,000 pounds. 
The test specimens were two inches long and turned to 0.625 
inch in diameter. 

At 38,000 pounds per square inch the gun bronze specimen 
was shortened about 41 per cent, of its original length, while 
its diameter had become 0.77 inch. 



Art. 43.] 



RESISTANCE OF ALLOYS. 



387 



TABLE II. 









2 


OF SHORT- 
AUSED BY 
T LOAD. 






COMPOSITION. 


POUNDS 
CAUSING 


PER SQUARE INCH 
A SHORTENING OF 


Q 2* 


HING : 
IN 1 

INCH. 


MANNER OF 








, K 


. U isi 


u> W 













CENT, 

ING 

EATE 


CRU 
TANC 
R SQ. 


FAILURE. 










Copper. 


Tin. 


S/'^ 


\o% 


20^ 


^ Z K 
OS |i! 

0. 


c. ^ ^ 

2 1/) a. 




97-83 


1.92 


29,340 • 


34,000 


46,000 


46,260 


0.37 


34,000 


Flattened, 


95.96 


3.80 


39,200 


42,050 


52,150 


52,150 


0.30 


42,050 


'^ 


92.07 


7-76 


3^500 


42,000 


65,000 


84,100 


0.45 


42,000 


ii 


90-43 


9-50 


32,000 


38,000 


60,000 


61,930 


0.34 


38,000 


ii 


87.15 


12.77 


39,000 


53iOoo 


80,000 


89,640 


0.39 


53,000 


ii 


80.99 


18.92 


65,000 


78,000 


103,490 


103,490 


0.20 


78,000 


ii 


76.60 


23 23 


101,040 






114,080 


0.09 


114,080 


Crushed. 


69.90 


29.85 








146,680 


0.04 


146,680 


ii 


65-31 


34.47 








84.750 


0.03 


84,750 


ii 


61.83 


37-74 









39i"o 


0.02 


39,iio 


'^ 


47.72 


51.99 









84.750 


0.02 


84,750 


ii 


44.62 


55-15 








35,850 


O.OI 


35,850 


ii 


38.83 


60.79 








39,110 


0.02 


39,110 


ii 


38.37 


61.32 








29,340 


O.OI 


29,340 


ti 


34.22 


65.80 


19060 






19,560 


0.06 


19,560 


ii 


25.12 


74.51 


17,930 


17,930 


17,930 


17,930 


0.28 


17,930 


ii 


20.21 


79.62 


16,300 


16,300 


16,300 


16,300 


0.29 


16,300 


^^ 


15.12 


84.58 


6,520 


6,520 


6,520 


9,450 


0.51 


6,520 


Flattened. 


11.48 


88.50 


10,100 


10,100 


10,100 


14,020 


0.50 


10,100 


** 


8.57 


91.39 


6,500 






9,780 


0.06 


9,780 


ii 


3-72 


96.31 


6,520 


6,520 


6,520 


9,780 


0.34 


9,780 


ii 


0.74 


99.02 


6,520 


6,520 


6,520 


9,780 


0.36 


9,780 


'^ 


0.32 


99.46 


6,520 


6,520 


6,520 


9,780 


0.38 


9,780 


ii 


Cast copper. 


26,000 


39,000 


51, coo 


74,970 


0.45 


39,000 


^^ 


ii bi 


33,000 


45,500 


58.670 


78,230 


0.43 


45,500 


** 


ii w 


34,000 


42,000 


58,000 


71,710 


0.32 


42,000 


'^ 


ic ii 


30,000 


36,000 


50,000 


104,300 


0.52 


36,000 


ii 


ii ii 


30,000 


37,000 


50,000 


91,270 


0.48 


37,000 


'^ 


li ii 


35,000 


48,000 


65,000 


97,790 


0.41 


48,000 


i i 


Cast tin. 


6,033 


6,400 


6,530 


7,500 


0.44 


6,400 





The copper specimen failed at 71,700 pounds per square 
inch, having been shortened about one-third of its length. 

The results of a series of tests by Prof. Thurston, in con- 
nection with the United States testing commission, are given 
in Table II. ; they were abstracted from " Mechanical and 
Physical Properties of the Copper-Tin Alloys," United States 
Report, edited by Prof. R. H. Thurston, 1879. ^^^ the speci- 
mens were 0.625 inch in diameter and 2 inches long. Scarcely 
one of them can be said to possess an elastic limit. 



388 



COPPER, ETC., IN COMPRESSION. 



[Art. 43 



The series of alloys presents some interesting results. 
About the middle third of the series are seen to be brittle 
compounds giving (as a rule) high ultimate compressive resist- 
ances, while the other two-thirds are ductile, and give at the 
copper end high results, and low ones at the tin end. 

It will be observed that Prof. Thurston took the load per 
square inch which gave a shortening of lo per cent, of the 
original length as the ultimate resistance to crushing of the 
ductile alloys and metals, since such materials cannot be said 
to completely fall under any pressure, but spread laterally and 
offer increased resistance. 

TABLE III. 



PER CENT. OF 


POUNDS PER SQUARE INCH FOR 


PER CENT. OF 
SHORTENING. 


MANNER 


Copper. 


Zinc. 


E\. 


Ult. Resist. 


OF FAILURE. 


96.07 
90.56 
89.80 

76.65 

60 94 

55-15 

49.66 

47-56 

25-77 

20.81 

14.19 
10.30 

4-35 
0.00 


3-79 

9.42 

10.06 

23.08 

33.65 

44-44 
50.14 
52.28 
73-45 
77.63 
85.10 
88.88 

94.59 
100.00 


305,500 
342,100 


29,000 

30,000 

29,500 

42,000 

75,000 

78,000 

117,400 

121,000 

110,822 

52,152 

48,892 

49,coo 

48,000 

22,000 


10. 
10. 
10. 
10. 
10. 
10. 
10. 

10. 

5.85 

2-75 
10.8 
10. 
10. 

TO.O 


Flattened. 


656,500 
1,772,500 


( < 
< ( 


1,345,500 
1,500,000 

4,232,800 

2,485,000 

897,000 


Crushed. 
Flattened. 




318,500 


(( 



Table III. contains the results of Prof. Thurston's tests of 
the copper-zinc alloys made while he was a member of the 
United States Board. The data are taken from " Ex. Doc. 23, 
House of Representatives, 46th Congress, 2d Session." The 
specimens were two inches long and 0.625 inch in diameter of 
circular cross section. 



Art. 44.] COMPRESSIVE RESISTANCE OF GLASS. 



389 



The values of E^ (ratios of stress over strain) are computed 
for about one-quarter the ultimate resistance. This ratio is so 
very variable for different intensities of stress that these alloys 
can scarcely be said to have a proper " elastic limit." 

In the ''Philosophical Transactions" for 1818, Rennie gives 
the following as the results of his experiments on 0.25 inch 
cubes : 



Fine yellow brass (10 per cent, shortening). 
Fine yellow brass (50 per cent, shortening). 
Cast lead (50 per cent, shortening). 



12,852 pounds per square inch. 

41,216 pounds per square inch. 

1,932 pounds per square inch. 



Art. 44. — Glass. 

The following results are taken from Sir Wm. Fairbairn's 
" Useful Information for Engineers," second series. The 
cylinders were about 0.75 inch in diameter and annealed. 

TABLE I. 



KIND OF GLASS. 


SPECIMEN. 


HEIGHT OF SPECIMEN. 


CRUSHING RESISTANCE, 
LBS. PER SQ. INCH. 


Flint 


Cylinder. 
( < 

< ( 
t < 

< ( 

( < 

< ( 
(( 

Cube. 

< ( 

i < 
(k 

<< 


Inch. 

I.OO 

I.OO 

1.60 
2.05 

I.OO 
1.50 
2.00 
I.OO 

1.50 

1. 16 
1. 10 
1. 10 

I.OO 
I.OO 

0.90 


23,480 
34,850 
20, 780 
32,800 
22,580 
35,030 
38,020 
23,180 
38,830 
14,240 
13,200 
13,260 
11,820 




(< 


Green 


(( 


Crown 


< ( 


Flint 


t k 




Green 


20,470 

I9'950 
21,870 


Crown 





390 



CEMENT, ETC., IN COMPRESSION. 



[Art. 45. 



It will be observed that the cubes give considerably less 
resistance than the cylinders. 

All the glass was annealed, but Fairbairn remarks that the 
cubes may have been only imperfectly so, since they were cut 
out of the interior of larger masses, while the cylinders were 
cut from rods as they were drawn. The latter, also, thus re- 
tained their natural skins, which may have increased their 
resistances. 

At the instant of failure the specimens were shattered into 
a great number of small pieces. 

Art. 45. — Cement — Cement Mortar — Concrete — Artificial Stones. 

Table I. of Article 37 contains the ultimate compressive 
resistances of a great number of pure cements, as tested by 
General Gillmore under the circumstances related in connec- 
tion with the table. The results are given in pounds per 
square inch. 

TABLE I. 



CEMENT, 


SAND, 
VOL. 


LENGTH, 
INS. 


DIAM., 

INS. 


MEAN 
OF 




THESE RESULTS ARE LBS. 
PER SQ. IN. 


AGE, IN 


VOL. 


Greatest. 


Mean. 


Least. 










I 
I 
I 
2 
2 
2 


T2 


2^.5 


114-j 

••I 


E. L. 

R. 

El. 
E. L. 

R. 

El. 
E. L. 

R. 

Ex. 


1,502 

1,889 

1,500,000 

587 
783 

1,910,000 
4^4 
489 

6,633,330 


800 

1,320 

800,000 

365 

494 
607,000 

182 

213 

1,285,000 


424 
620 
500,000 
191 
261 

217,333 
98 

131 

220,450 


/123 to 

j 143- 

I117 to 

\ 141- 

/ 127 to 

f 135. 



Table I. contains the results of tests of ** Fall City " (Louis- 
ville) cement and cement mortar. The tests were made by 
Mr. Bremermann, by direction of Capt. Eads, during the con- 



Art. 45.] 



CEMENT MORTARS. 



391 



struction of the St. Louis bridge. " E. L." is the elastic Hmit ; 
" R." the ultimate resistance to compression ; and '* E/' the co- 
efficient of compressive elasticity, all in pounds per square 
inch. 

The foUowinsf results are from the same source: 











Akron Cement. 










4 


24- 


-inch cubes ; 


I vol. 


cement 


, vol. 


sand ; 


iilL resist 


= 


2 


,140 lbs 


2 






2 " 




T " 




i ( < ( 


= 


I 


,105 " 


3 






I 




I " 




< ( < ( 


= 




733 " 


2 






2 ' ' 




3 " 




" <' 


= 




520 ** 


I 






I " 




2 " 




< < ( ( 


= 




240 " 


I 






I " 




4 " 




( < ( ( 


= 




480 " 



" EaH City,'' Louisville, Ceiiient. 
3 2^-inch. cubes ; i vol. cement, o vol. sand ; tilt, resist. 



1,587 lbs. 
640 " 
4C0 " 
240 ** 



Louisville Cement fro7?i Beach ^ Co. 

4 2^-inch cubes ; i vol. cement, o vol. sand ; ult. resist. = 1,615 lbs. 

I " " " 2 " " I " " " " = 1,280 " 

1 " " I " " " " = 560 " 

2 " " 3 " " " " — 400 •' 
I " "2 " " " " — 280 " 



I 


< ( 


( t 


I 


( ( 


t ( 


2 


< ( 


(( 



Louisville Cement from Hubne ^ Co. 

2 2i-inch cubes ; i vol. cement, o vol. sand ; ult. resist. = 2,320 lbs. 

2 " " " I " '♦ I " " " " = 740 " 

I " " " 2 " " 3 " " " •' = 600 " 



These ultimate resistances are in pounds per square inch. 
All specimens were " moulded under hand pressure only, 
left 12 days in water, and exposed six months to the air." 



392 



CEMENT, ETC., IiV COMPRESSION. 



[Art. 45, 



TABLE \a. 



DESCRIPTION OF PORTLAND CEMENT AND MORTAR. 



Neat Portland cement , 

I Portland cement to i pit sand. 

2 " . 
<« _ < ( 

4 

5 " . 

Neat Portland cement 

I Portland cement to I sand. . . . 

" 2 " 

<< 'i " 

4 

5 " .-.. 

Neat Portland cement 

I Portland cement to i pit sand. 

2 " . 

3 " . 

4 *' . 

5 " . 



ULT. RESIST. IN 
POUNDS PER. SQ. IN. 



3,795 
2,491 
2,004 
1,436 
1,331 
959 
5,388 

3.478 
2,752 
2,156 
1,797 
1,540 
5,984 
4,561 
3,647 
2,393 
2,208 
1,678 



The results of a large number of experiments on the com- 
pressive resistance of Portland cement and mortar, at different 
ages, by Mr. John Grant, C. E. ('' On the Strength of Cement," 
1875), are given in Table \a. The specimens were made into 
bricks 9 X 4.25 X 2.75 inches, and were compressed on their 
flat sides of 9 X 4.25 = 38.25 square inches area. The results 
are in pounds per square inch. 

Table II. is taken from the same work, and shows the cir- 
cumstances under which Mr. Grant made his experiments. 
Two sets of blocks were made in each case ; one set was kept 
in air for one year, and the other in water for the same length 
of time. The cubes were then crushed with the results shown. 
It is to be observed that the results are pounds per square 
foot. Two series of the blocks were formed by compressing 
the material in layers one inch thick ; the others were not 
compressed. 



Art. 45.] MORTARS AND ARTIFICIAL STONES. 



393 



TABLE II. 



fe h.' 

H g 

fc J " 
2! u S 


ULTIMATE COMPRESSIVE RESISTANCE IN POUNDS PER SQUARE FOOT. 


DF BALLi 
^D GRAY 
.. OF CE 


Compressed. 


Not Compressed. 


1 a ^ 
ail 




Kept in 




Kept in 




Kept in 


0^0 


Kept in air. 




Kept in air. 




Kept in air. 




> 




water. 




water. 




water. 




(Exceptional.) 












I 


239,680 


381,920 


340,480 


301,060 


268,800 


336,000 


2 


333,760 


358,400 


385,280 


309,120 


344,960 


322,560 


3 


253,120 


258,720 


268,800 


318,080 


215,040 


250,880 


4 


230,720 


243,040 


268,800 


250,880 


250,880 


241,920 


5 


199,360 


222,880 


219,520 


318,080 


215,040 


210,560 


6 


180,320 


203,840 


182,784 


175,616 


163,072 


152,320 


7 


168,000 


180,320 


147,840 


143,360 


125,440 


112,000 


8 


137,760 


170,240 


120,960 


120,960 


112,000 


98,560 


9 


120,960 


153,440 


107,520 


98,560 


89,600 


80,640 


10 


108,640 


107,520 


94,080 


94,080 


71,680 


62,720 




12" X 12" X 


12" blocks. 


6"x 6"x 


6" blocks. 


6" X 6" X 6 


" blocks. 



Table Wa. is from the same source as Table I. 

The concrete blocks were pressed evenly on 36 square 
inches until failure took place. 

The following results for artificial stones are given by Mr. 
Henry Reid ('' A Practical Treatise on Natural and Artificial 
Concrete," 1879) • 

A 4-inch cube of Ransome's " Siliceous Stone " gave 4,200 
pounds per square inch. 

By experimenting with 2-Inch cubes of "Rock Concrete'* 
pipes, Mr. Reid obtained the following results from two series : 



GREATEST. MEAN. LEAST. 

4,340 3,454 2,684 pounds per sq. in. 

5,650 4,428 3,401 " " " '' 

5.650 4,763 3,107 •' " •' " 



394 



CEMENT, ETC., IN COMPRESSION [Art. 45. 



TABLE Ila. 

6" X 6" X 6" Concrete Blocks. 



^ 






VOLS. AKRON 


VOLS. LOUIS- 


VOLS. BROKEN 


ULT. RESIST., LBS. 


xn 




VOLS. SAND. 

















CEMENT. 


VILLE CEMENT. 


LIMESTONE. 


PER SQ. IN. 


X 
4) 














C 














^ 




I 







4 


889 


Ti 




I 







4 


1,124 






I 







4 


1,170 






2 







4 


722 


t/3 




2 







4 


889 




2 







4 


1,361 






I 





I 


4 


1,194 


u 




2 





I 


4 


950 


c3 




2 





I 


4 


640 


^ 




2 





I 


4 


890 






I 


0-5 


0-5 


4 


1,170 




in 


I 


0-5 


0-5 


4 


1,361 


;=i 


"S 


I 


0.5 


0.5 


4 


1,445 







a 


I 


0.5 


0.5 


4 


1,280 



^ 


vO 


I 


0-5 


0.5 


4 


1,250 


-3 


1-1 




2 


0.5 


0.5 


4 


918 


<tH 


2 


0-5 


0.5 


4 


1, 000 







2 


0.5 


0.5 • 


4 


1,140 





2 


0.5 


• 0.5 


4 


1,361 


r— 1 


rg 


2 


0.5 


0.5 


4 


611 




^ 


2 


0.5 


0.5 


4 


I, III 



There were six experiments in each series. 
Three-inch cubes of Victoria stone (six experiments in the 
first series, and ten in the second) gave : 



GREATEST. 



5,179 4,422 3,294 pounds per sq. in. 

4,70s 3,955 3,578 pounds per sq. in. 

These cubes were made in February, 1879, ^^^ broken in 
May of the same year. 

*' Two-inch cubes of siHcated stone made with 3 parts 
Thames ballast and i of Portland cement, gauged with water, 
and put in the silicate bath for 11 days, about 12 months old 
fractured as follows : 



Art. 45.] ARTIFICIAL STONES. 395 

No. 1 4>237 pounds per square inch. 

No. 2 5,650 pounds per square inch. " 

Four "granitic breccia" cubes, 3" x 3", about 25 years old, 
gave the following results : 

GREATEST. MEAN. LEAST. 

8,886 8,028 7,533 pounds per square inch. 

Seven blocks of Sorel stone, varying from ij^ x i^^ x i 
inch to 2 X 2y^ X ij^ inches gave: 



ij. 



AGE. INERT MATERIAL. ULT. RESIST. 

I year Coral sand 6,240 lbs. per sq. in. 

1 " Pulverized quartz 7,270 " " " 

2 years Washed flour of emery 19,640 " " " 

3 " Fine marble 11,560 '* " " 

9 months Mill sweepings 6,130 " " " 

2 years Marble and sand 4, 920 " " " 

Not known Marble with colored veneer 7,680 " " " 

The weight of the oxide of magnesium varied from 12 to 
15 per cent, of the whole. 

The results of a series of tests, by Gen. Gillmore, in 1870 
and 1871, on coignet b^ton blocks, 3.5 x 5.5 x 3 inches, are 
given in Table III. Two blocks of each kind were tested. All 
the blocks were two months old. The results are in pounds 
per square inch. 

With four 2-inch cubes of Frear stone. Gen. Gillmore ob- 
tained the following results : 

Four weeks old 4>500 pounds per square inch. 

Four weeks old 4,626 " " " 

Three v^'eeks old 2,250 " " " " 

Six months old 2,000 " " " '' 

These blocks were composed of one measure of hydraulic 
cement, two and a half of sand, moistened with an alkaline 
solution of gum shellac of sufficient strength to furnish one 



39^ 



BRICKS IN COMPRESSION. 



[Art. 46. 



TABLE III. 

















COMP. RESIST. Iff 




PROPORTIONS 


BY 


VOLUME, 


LOOSELY MEASURED. 


















POUNDS PER SQ. IN. 




Cement, i 


common 


lime 


powder, 0.4; sand, 5.6.. 




935 

831 




" I 










0.8; " 5.6.. 




805 

987 














0.4; " 7.5.. 




416 
. 519 















0.8; '• 7.5.. 




551 


PL, 












1 i ' %J ' ' 




571 


f3 












» j 0.4; " 5.6[ 
gravel and pebbles, 5 j 




649 














681 


.2 
'9 












a i 0-4 ; sand, 5.6 ) 
gravel and pebbles, 13 j" 




675 

















831 














«. _^ 0.8 ; sand, 5.6 ) 
i gravel and pebbles, 5 ) 




649 














I 623 




<. 










a i 0.8 ; sand, 5.6) 
( gravel and pebbles, 13 f 




649 














. 753 



ounce of the shellac to i cubic foot of the finished stone. 
Portland cement was used in the first three blocks and Louis- 
ville cement in the last. 

Specimens of artificial stone made under the Van Derburgh 
system and used in the walls of the Howard University and 
Hospital buildings at Washington, D. C, in 1868 and 1869, 
varying in age from 3 to 16 months, gave resistances of 173 (4 
months old) to 564 (10 months old) pounds per square inch. 
Another specimen of the same stone, ten years old, gave 1,45 5 
pounds per square inch. 

Art. 46. — Bricks. 



The first set of results given below are computed from data 
given by Gen. George S. Greene, Jr., C. E., in Vol. II. of the 
" Trans. Am. Soc. of Civ. Engrs." 



Art. 46.] 



RESISTANCE OF BRICKS. 



397 



Nos. i» 2 and 4 cracked, but did not crush to pieces, as the 
others did. 



NO. SIZE OF BRICK. 

Inches. 
1 2.3 X 3.52 X 4.4 . 

2 2.24 X 3.5 X 4.46. 

3 2.34 X 3.5 X 4.52. 

4 2.34 X 3.46 X 4.46. 

5 2.30 X 3.46 X 4 

6 2.28 X 3.46 X 4 



50. 
60. 



Sq. ins. 



COMP. RESIST. 



Lbs. per sq. in. 



15 
15 
15 
15 
15 
15 



5 3,230 

6 3,360 

8 2,750 

4 1,994 

6 2,050 

9 2,920 



The pressure was applied on the two opposite largest faces 
of the bricks, giving blocks whose heights were only 0.7 their 
least widths. 

In Vol. VII. of the " Trans. Am. Soc. of Civ. Engrs." Mr. 
Francis Collingwood, C. E., gives the following as the results 
of compressing ten whole bricks 07i end : 

GREATEST. MEAN. LEAST. 

.3,060 2,065 1,524 pounds per square inch. 

For ten half bricks on small side : 

6,400 4,610 2,900 pounds per square inch. 

For ten half bricks on flat side : 

4,150 3,370 2,670 pounds per square inch. 



In regard to these tests Mr. Collingwood says, " The 
bricks were selected to give a fair average of ' good Haver- 
straw stock brick,' not the hardest burned. No packing was 
inserted in the machine between the bricks and the com- 
pressing surfaces ; so that the strength in compression repre- 
sents the case of imperfect beds, etc., although it was found 
that it made but little difference." He attributes the higher 
values for the ** ten half bricks on small sides," over those be- 



398 



NA TURAL STONES IN COMPRESSION. [Art. 47. 



longing to the '* half bricks on flat side," to the imperfect 
bearing surfaces of the latter. 

Art. 47. — Natural Building Stones. 

The ultimate resistances and coefficients of elasticity given 
in Table I. were determined in connection with the construc- 

TABLE I. 



MATERIAL. 



Grafton Majznesian limestone 



Portland granite 



Richmond " . . . 

Portland 

Missouri red granite 



LENGTH IN 
INCHES. 



6.46 

5.87 
5-96 

5-99 
3.00 

8.00 
[3.00 
5.88 
5-98 
5-97 
6.00 
3.00 
3.00 
3.00 
3.00 
3.00 



DIAM. IN 



I. 14 
1.06 
1.06 
I 07 

2.38 

I-I3 
2.36 
2.36 
2. 38 
2.30 
3x3 

3x3 
3x3 
3x3 
% -X. '\ 



POUNDS PER SQ. IN. FOR 



Ultimate 
Resistance. 



7,200 
8.500 
2 000 
6,000 
15,400 
10,100 
10 800 
16,000 
18,500 
17,000 
16,400 
13,700 
12,700 
13,000 
12,700 
13,600 



10,500,000 
8,400,000 
8,500,000 
6,000,000 



12,000,000 
5,000,000 
5,500,000 
6,400,000 
5,000,000 

13,500,000 



tion of the St. Louis arch, and have been taken from Prof. 
Woodward's history. The following results for Missouri stones 
are from the same source : 



ULT. RESIST. 

3" X 3" X 3" cube brown ochre marble 15.000 lbs. per sq. in. 

3" X 3" X 3" " sandstone from Ste. Genevieve. . 5, 33© " " " " 

4I X 4^ X 4| '• " " '' " .. 5.500 -■ -- " " 

3rV X 3iV X zh " " " " " •• 3,4co '' " " " 



Art. 47.] 



BUILDING STONES. 



399 



TABLE II. 

Two-inch Cubes. 



Blue 

Dark 

Light 

Flagging . . 
Old Quarry., 
Old Quarry.. 

Up River 

Up River 
Niantic River 
Niantic River 
Porter's Rock 
Porter's Rock 

Gray 

Gray 

Gray 

Gray 

Gray 

Gray 

Gneiss 

Gneiss 

Dark 

Dark 

Bluish-gray .. 

Gray 

Glen's Falls. . 
Glen's Falls. . 

Lake 

Lake 

North River. 
North River. 

White 

White 

Drab 

Drab 

Drab 

Drab 

Dark 

Dark 

Drab 

Drab 

Caen 

Caen 

East Chester. 
East Chester. 
Vermont . ... 
Vermont 



LOCALITY. 



Staten Island, N.Y... 

Di.K Island, Me 

Quincy, Mass 

Quincy, Mass 

North River 

Westerly, R. I 

Westerly, R. I 

Richmond, Va 

Richmond. Va . 

New London, Conn . . . 
New London, Conn .. . 
Mystic River, Conn ... 
Mystic River, Conn ... 

Westerly, R. I 

Westerly, R. I 

Richmond, Va 

Richmond, Va 

New Haven, Conn 

New Haven, Conn 

Sachemshead Quarry, 

Conn 

Sachemshead Quarry, 

Conn 

DuldLh, Minn. 

Huron Island, Mich... 

Keene, N. H 

Pompton, N.J 

Glen's Falls, N. Y 

Glen's Falls, N. Y.. .. 
Lake Champlain, N. Y. 
Lake Champlain, N. Y. 

Kingston, N. Y 

Kingston. N. Y 

Jolict, III 

Joliet, 111 

Lime Island. Mich 

Lime Island. Mich 

Marquette, Mich 

Marquette, Mich 

Bardsiown, Ky 

Bardstown, Ky 

Canton, Mo 

Canton, Mo 

France 

France 

Tuckahoe, N. Y 

Tuckahoe. N. Y .. 

Dorset. Vt 

Dorset, Vt 



O 








IT. 


u 




ca 


K 




K 


^ 


• 






z 


(- y 









^ 






(ij H 


h 


w 


« 







s - 






2 




s &. 


D 












u 


0, 


Bed. 


22.250 


178 8 




15,000 


166. 5 




17^750 


166,2 




M-750 


168.7 




13.425 


168.1 




•^l-lb° 


165,6 




17,250 


165,6 




21.250 






20.000 






12.500 


166.3 


Edge 


14-175 


166.3 


Bed. 


18,125 


164.4 


Edge. 


22.250 


164,4 


Bed. 


14.687 


166.9 


Edge. 


I4i937 


166,9 


Bed. 


14.100 


164.4 


Bed. 


13-875 


164.4 


Edge. 


7.750 


162,5 


Bed. 


9500 


162.5 


Edge. 


IS-Q37 


163.7 


Bed. 


14.000 


163.7 


Bed. 


17-750 


173-7 


Bed. 


18,125 


164,4 


Bed. 


10,375 


166,0 


Bed. 


24,040 




Bed. 


11.475 


168.8 


Edge. 


10,750 


168.8 


Bed. 


25,000 


171.9 


Edge. 


21.500 


171. 9 


Bed. 


13.9CO 


168.2 


Edge. 


11.C50 


168.2 


Bed. 


12-775 


158.8 


Bed. 


:6,c;oo 


162.5 


Bed. 


25.000 


161.2 


Bed. 


'5-425 


159-4 


Bed. 


7-825 


146.3 


Edge. 


7 600 


146.3 


Bed 


16.250 


166.9 


Edge. 


15.000 


166.9 


Bed. 


9.250 


146,0 


Bed. 


5650 


146.0 


Bed. 


3650 


118.8 


Bed. 


3-450 


118.8 


Bed. 


12.050 


179.7 


Bed. 


12.050 


179-7 


Bed. 


7.612 


164.7 


Bed. 


8,670 


167.8 



REMARKS. 



Cracked before bursting. 
Burst suddenly. 
Cracked before bursting. 
Cracked before bursting. 
Broke suddenly. 



Waxy-looking. 
Broke suddenly. 



Syenitic. 



Average of 3. 

Burst without cracking. 



Rather a clay stone. 



400 



NATURAL STONES IN COMPRESSION. [Art. 47. 



TABLE \\—Contimied. 



Drab 

Drab 

Drab 

Drab 

Common Ital. 
Common Ital. 

Brown 

Brown 

Gray 

Gray 

Brown 

Brown 

Pink 

Pink 

Drab 

Drab 

Drab 

Drab 

Drab 

Purple 

Purple 

Purple 

Purple 

Red-brown . . 
Red-brown . . 
Olive green . . 
Olive green. . 

Brown 

Brown 

Pink 

Pink 

Light buff . . 
Light buff. . . 
Freestone. . . . 

Freestone 

Yellow drab. 
Yellow drab. 
Craigleith . . 
Craigleith 



Mill Creek Quarry, 111. 

Mill Creek Quarry, 111. 

North Bay Quarry, Wis. 

North Bay Quarry, Wis. 

Italy 

Italy 

Little Falls, N. Y 

Little Falls, N. Y 

Belleville, N. J 

Belleville, N. J 

Middletown, Conn .... 

Middletown. Conn .... 

Medina, N. Y 

Medina, NY 

Berea, Ohio 

Berea, Ohio 

Berea, Ohio 

Vermillion, Ohio 

Vermillion, Ohio 

Fond du Lac, Wis . . . . 

Fond du Lac. Wis 

Marquette, Mich 

Marquette, Mich 

Seneca Freestone, O. , . 

Seneca Freestone, O. .. 

Cleveland, Ohio 

Cleveland, Ohio 

Albion, NY 

Albion, N. Y 

Kasota, Minn 

Kasota, Minn 

Fontenac, Minn 

Fontenac, Minn 

Dorch'ter, New Bruns- 
wick . . 

Dorch'ter, New Bruns- 
wick 

Massillon, Ohio 

Massillon, Ohio . 

Edinburgh, Scotland . . 

Edinburgh, Scotland.. 



2 



Bed. 

Edge 

Bed 

Edge. 

Bed. 

Bed 

Bed. 

Edge. 

Bed. 

Edge. 

Bed. 

Edge 

Bed 

Edge. 

Bed. 

Bed. 

Bed. 

Bed 

Bed. 

Bed. 

Edge. 

Bed. 

Edge. 

Bed. 

Edge. 

Bed. 

Edge. 

Bed. 

Edge. 

Bed. 

Edge. 

Bed. 

Edge. 

Bed. 

Edge. 
Bed. 
Bed. 
Bed. 

Edcre. 



687 

787 

025 

,700 

250 

,062 

850 

,150 

11.7CX) 

10,250 

6.950 

5r5.50 

17,250 
14.812 
10,250 

8,300 
250 
.250 

000 



250 

ilIO 

450 
.730 
,687 

,500 

,800 

,910 

,500 
.350 

,700 

,675 

,250 
>775 



9-150 

6.050 
8.750 

6:725 

12.000 
11,250 



160. 

156. 

175 

175- 

163. 

168. 

140. 

140. 

141. 

141. 

148. 

148. 

150. 

149. 

131- 

133- 

137- 

135 

135. 

138. 

138. 

^35 

135. 

149. 

149. 

140. 

140. 

151. 

151- 

164 

164. 

145 

145. 



131.8 
131.8 
M1..3 
141-3 



Broke sud'ly. Hardened 
by years of exposure. 



Calcareous. 



Table II. and the other tables of this article contain the re- 
sults of tests given in the " Report on the Compressive Strength, 
Specific Gravity and Ratio of Absorption of the Building 
Stones in the United States," by Gen. Q. A. Gillmore, 1876. 



Art. 47.] 



SANDSTONE CUBES. 



40 T 



The specimens, whose tests are given in Table II., were 2-inch 
cubes. " Each cube was placed between two cushion blocks of 
soft pine wood, 2 inches by 2 inches square, and slightly more 
than 0.25 inch in thickness ; one on the top and the other 
under the bottom ; the grain of the wood being parallel in each 
to the other — though no difference was observed when this 
was changed, as regards amount of record." . . . *' The 
cubes were brought to a true, smooth and regu- 
lar, but not a polished surface." The third column shows 
whether the specimen was crushed " on bed " or " on edge." 

TABLE in. 

Berea Sandstone Cubes. 





COMP. RESIST., LBS. PER 




COMP. RESIST., LBS. PER 


EDGE OF CUBE. 




EDGE OF CUBE. 






SQUARE INCH. 




SQUARE INCH. 


Inch. 


Pounds. 


Inches. 


Pounds. 


0.25 


4,992 


2.00 


8,955 


0.50 


6,080 


2.25 


9>i30 


0.75 


6,347 


2. 50 


8,856 


I. 00 


6,990 


2.75 


9,838 


1.25 


7>342 


3.00 


10,125 


1.50 


8,226 


4.00 


11,720 


1-75 


9>3io 











General Gillmore showed that the size of the cube tested, 

affected very greatly the ultimate compressive resistance per 

unit of area of face of cube. Table III. shows the results of 

gradually increasing the size of cubes of Berea sandstone, 

crushed " on bed " between wooden cushion blocks increasing 

(with size of cube) from about 0.0625 inch to about 0.4 inch in 
26 



402 



NATURAL STONES IN COMPRESSION. [Art. 47. 



thickness. The general result is very marked in spite of two 
or three irregularities. 

These results are natural consequences of the character of 
stone and the cubical form of the specimens. A few of Gen- 
eral Gillmore's experiments showed that such results would 
probably not appear if the length of the specimens had been 
two or three times the width or breadth. 

The effect of different bearing surfaces on the ultimate 
compressive resistance of stone cubes is well shown by the 
results given in Table IV. All the results are in pounds per 
square inch, and belong to two-inch cubes, with the exception 
of the "Sandstone, drab" specimens, which were 1.5 inch 
cubes. Each result is a mean of two to five tests. 

TABLE IV. 





ULT. COMP. RE.SI.ST., POUNDS PER SQl'AKE INCH. 




Steel. 


Wood. 


Lead. 


Leather. 


Granite, Millstone Point, Conn. . . . 
Granite Keene N. H 


23,190 
24,000 

19,125 

11,260 
13,280 

1,075 

4,000 
8,500 
5,660 


22,880 
19,830 

^7,540 
10,290 
10,850 

1,075 
4,000 

8,750 
6,730 


15,730 
14.480 
11.560 
7.380 
9,200 
1,075 
4.000 
7,250 
5,500 




15,730 


Marble, East Chester, N. Y 

Sandstone, Berea Ohio 


6,730 
8,190 

1,075 


Vermont marble, Vt 


Limestone, Sebastopol 




Sandstone, Massillon, Ohio 

Sandstone, Massillon, Ohio (softer). 




3,640 



The steel cushion gave the highest results by a little. A 
soft cushion seems to be driven into the small cavities and in- 
terstices of the specimen, and thus to produce a splitting 
action at the bearing surfaces. ** The beds of the granite and 
marble cubes were rubbed to the border of polish ; those of 
sandstone were rubbed smooth." 



Art. 48.] 



TIMBER. 



403 



Again, polished and unpolished cubes give different resist- 
ances per square inch, as shown in Table V. The results there 
given are for two-inch cubes pressed upon by wooden cushions. 

It is at once evident that the polished cubes gave consid- 
erably the highest resistances. This is probably due to the 
fact that the splitting action of the wooden cushions was re- 
duced to a minimum on the polished surfaces. 

TABLE V. 



KIND OF STONE. 


ULT. COMP. RESIST., TER SQUARE INXH. 


Polished. 


Unpolished. 


Granite, Quincy, Mass 


Pounds. 
24.750 
25,000 
21,630 
23.750 
22,880 
19,830 
23.500 
17.540 
10,850 


Pounds. 
17.750 
22,250 
13,380 
18,250 
18,750 
12,750 
17,750 
12.950 
8,750 


Granite, Staten Island, N. Y 


Granite, Garrison's, N. Y 


Granite, Tarrytown. N. Y 

Granite, Millstone Point, Conn 

Granite, Keene, N. H 


Granite, Westerly, R. I 


Marble, East Chester, N. Y 


Marble, Vermont, Vt 





General Gillmore's experiments show, in a very conclusive 
manner, that variety in circumstances of testing will produce 
a variety of results for the same section of stone specimen. 
Attending circumstances and dimensions of specimens, there- 
fore, should always be given. 



Art. 48.— Timber. 

Table I. is based upon results of experiments made at the 
Stevens Institute, which were given by Prof. Thurston in the 
Journal of the Franklin Institute for Oct., 1879. The speci- 
mens were well seasoned and turned to about 1. 125 inches in 



404 



TIMBER IN COMPRESSION. 



[Art. 48. 



TABLE I. 



WOOD. 


POUNDS OF STRESS PER SQUARE INCH AT 


PER CENT. OF 


Ult. Resist. 


Elas. Lim. 


Coefficient of Elas. 


FINAL SHORTENING. 


White pine 

Yellow pine 

Locust 


9' 590 

11,950 

14,820 

7,000 

8,150 

7,140 

10,410 


5,600 
7,000 
9,800 
5,700 
5,180 
5,600 
6,300 


354,400 
469,800 
604,950 
1,079,500 
713,300 
361,300 
594.350 


3-5 

2.9 

3-3 
1.25 

2.3 
3.3 
3.4 


Black walnut. . . . 

White ash 

White oak 

Live oak 



TABLE II. 





NO. OF EX- 
PERIMENTS. 


ULT. RESIST. IN POUNDS PER SQUARE INCH. 




Greatest. 


Mean. 


Least. 


Georgia pine 


9 
9 
9 

9 
9 
9 


11,500 
7,500 

12,580 
9,780 
8,410 
6,280 


9>520 
6,640 
11,720 
8,000 
7,860 
5,690 


8,170 
5,880 
11,010 
6,530 
7,170 
5,210 


^Vhile pine 


Locust 

^Vhite oak 


Spruce 


Hemlock 





Art. 48.] 



LASLETT'S EXPERIMENTS. 



405 



diameter with a length of 2.25 inches ; they were compressed 
in the direction of the fibre. The coefficients of elasticity were 
computed at the *' elastic limit," i. e., at the point at which 
permanent set began. 

Table II. contains the results of experiments made by 
R. G. Hatfield ("Transverse Strains," 1877). The specimens 
were from one to two diameters high, and were compressed in 
the direction of the fibres. 

The mean results of numerous English experiments by 
Thomas Laslett ('* Timber and Timber Trees, Native and 
Foreign," 1875) are given in Table III. He found very little 
difference in the results for i-inch, 2-inch, 3-inch and 4-inch 
cubes ; those for the smaller cubes, as a rule, gave a slight 
excess over the others. The cubes were crushed in the direc- 
tion of the fibre. 

TABLE III. 



TIMBER, 

I, 2, 3 and 4-inch Cubes. 


ULT. RESIST. IN 
LBS. I'ER SQ. IN. 


TIMBER, 

I, 2, 3 and 4-inch Cubes. 


ULT. RESIST. IN 
LBS. FER SQ. IN. 


Oak, Engflish (unseasoned). 
Oak, English (seasoned) ... 

Oak, French 

Oak, Tuscan 

Oak, Sardinian 


4,900 
7.480 
7.950 

5. 470 

5,835 

7.480 

6,070 

5.890 

5.730 

11,670 

12,590 

14,420 

8,470 

6,400 

6,-j8o 
5,690 
4,210 


Mahogany, Mexican 

Eucalyptus, Tewart 


5,600 
9'350 
7,170 
10,300 
6,900 
6,970 
5,490 


Eucalyptus, mahogany 

Eucalyptus, iron-bark 

Eucalyptus, blue-gum 

i Ash, English 


Oak, Dantzic 


Oak, Amciican, white 


i Ash, Canadian 


Oak, American, Baltimore.. 


Elm, English 


5,780 


Teak, Moulmein 


Elm, rock 

Hornbeam 


8,580 
8,310 
6,940 
5.240 
4,850 


Iron wood 


Chow 


Fir, Dantzic 

Fir. Riga 


Greenheart 


Sabicu 


Fir, spruce 


Mahogany, Spanish 


Larch 


5,820 
4,480 
6,470 
6,430 


Mahog^any, Honduras 

Red pine 


Cedar 


Pitch pine 


Yellow pine 


Kauri 







The results of the compressive tests of short blocks of 
timber, during the construction of the St. Louis bridge, are 
given in Table IV. These are especially valuable, both in 



4o6 



TIMBER IN COMPRESSION. 



[Art. 48. 



consequence of the large size of the blocks and the fact that 
the pressure was applied with and across the fibre. 

The blocks are seen to be from two to eight times as strong 
with the fibre as across it. 

TABLE IV. 



KIND OF TIMBER. 


WITH OR PER- 
PENDICULAR 
TO FIBRE. 


DI.MENSIONS 
I.V INCHES. 




< 

S 


f LTI.MATE RESISTANCE IN 
FOUNDS PER .SQ. IN. 


REMARKS. 




Greatest. 


Mean. 


Least. 




White oak 

White oak 

Black oak 

Gum 


Perp. 

With. ■ 

Perp. 

Perp. 

Perp. 

Perp. 

Perp. 

With. 

Perp. 

With. 

Perp. 

With. 

With. 

With. 

With. 


4x4x4 
4x4x4 
3x3X3 
3x3x3 
3x3x3 
3x3x3 

6x6x6 
6x6x6 
6x6x6 
6x6x6 
6x6x6 
6x6x6 
6x6x6 
6x6x6 
6x6x6 


4 
4 
2 
2 
2 
2 
3 
3 
3 
3 
3 
3 
2 
2 
2 


2,200 

2,000 
2,700 

440 
3.KJ0 

722 

1,222 

4,9'7 

4-14 

3. if 6 

3.^94 
4,7-^2 
3^778 


1,750 
3vV5 
i,8oo 
2,250 
33.5 

6ro 
3,241 
1,092 

4,79^ 
426 

3, "I 
3,^91 
4,611 

3,764 


i,3f5^ 
3,200 
1 ,600 
i.Soo 

330 
2,000 

5t5 
3,»'^3 
1,000 
4,722 

417 
3.000 
2,889 
4,500 
3,750 


1 Not well 
f seasoned. 

1 


Cvoress 




Ash 


•d 


White pine 

White pine 

Yellow pine 

Yellow pine 

Cypress 


c 


m 

• u 
1 « 


Cypress 


J 


White pine 

Yellow pine ..... 
White oak 



Table V. contains the results of tests by Colonel Laidley, 
U.S.A., ** Ex. Doc. No. 12, 47th Congress, 2d Session.** A 
few other tests of short blocks from the same source will be 
found in the article on '* Timber Columns." Unless otherwise 
stated, all the specimens were thoroughly seasoned. 

In this table, the " length " of all those pieces which were 
compressed in a direction perpendicular to the grain might, 
with greater propriety, be called the thickness, since it is 
measured across the grain. 

In the tests (24-60), the compressing force was distributed 
over only a portion of the face of the block on which it was 
applied ; thus the compressed area was supported, on the face 
of application, by material about it carrying no pressure. In 
some cases, this rectangular compressed area extended across 



Art. 48.] 



LAIDLEY'S EXPERIMENTS. 



407 



the block in one direction but not in the other. In all such 
instances the ultimate resistance was a little less than in those 
in which the area of compression was supported on all its 
sides. 

TABLE V. 







•J-. 
5 


COMPRESSED 


ILT. KESIST., 










s 






PEKP. TO OK 






'A 


KIND OF WOOD. 



u 

.4 


SECTION IN 
INCHES. 


LBS. PER SQ. 
INCH. 


WITH GKAIN. 


REMARKS. 


I 


Oregon pine 


16.5 


2.46 X 2.0 


8,496 


With. 




2 


Oregon pine 


19.9 


1.21 X 1.2t 


9,041 






3 


Oregon pme ...... 


19.9 


I. 21 X I. 21 


8,253 


•* 




4 


Oregon maple 


8.0 


3.6J X 3.6j 


6,66 1 


'' 




5 


Oregon spruce 


24.02 


3-9^ ^ 5.75 


5,772 


*' 


Unseasoned. 


6 


Calilornia laurel .. . 


8.0 


3.53 X 3-58 


6,734 


" 


Worm-eaten. 


7 


Ava Mexicana. . . 


8.0 


3.69 X 3.69 


6,382 


u 




8 


Oregon ash 


8.0 


3.64 X 3.64 


5,121 


11 




9 


Mexican white ma- 














hogany.. 


8.0 


3-77 X 3-77 


6,155 


" 




10 


Mexican cedar 


8.0 


3-75 X 3.75 


4,814 


" 




II 


Mexican mahogany 


8.0 


3.75 X 3-75 


10,043 


" 




12 


White maple 


12.0 


4.00 X 4.00 


7,140 


»k 




13 


White maple 


1'2.0 


4.00 X 4.00 


7,210 


" 




14 


Red birch 


13.0 


4.26 X 4.26 


8,030 


"■ 




15 


Red birch 


130 


4.26 X 4.26 


7,820 


" 




16 


Whitewood 


12.0 


4.00 X 4.00 


4,440 


" 




17 


Whitewood 


12.0 


4.00 X 4.00 


4,33c' 


kk 




18 


White pme 


12.0 


4.00 X 4.00 
4.00 X 4.00 


5,475 
5,7^0 


11 




19 


White pine 


12.0 


ki 




20 


White oak 


12.0 
J2.0 
12 .0 


4 . 00 X 4 . 00 
4.CX) X 4.00 
4.00 :< 4.00 
4.00 X 4.00 


7,373 
7,010 

7-940 
7,640 


Ik 




21 


White oak 




22 


Ash 




23 


Ash 


12.0 


" 




24 


Oregon pine 


1.05 


3.45 X 3.00 


1,150 


Pcrp. 




25 


Oregon maple 


Z.(^}, 


3.63 X 3. a? 


1,875 


" 




26 


Oregon spruce. .... 


3-92 


5-75 X 4-75 


710 


" 


Unseasoned, 


27 


Oregon spruce. . . . 


3-92 


4.75 X 4.00 


680 


" 


Unseasoned. 


28 


California laurel. . . 


3.58 


3.58 X 3.cx> 


2,000 






29 


Ava Mexicana 


3-^9 


3.69 X 3.00 


2,100 






30 


Oregon ash 


3.64 


3.64 X 3.00 


2,200 


" 




■ 31 


Mexican white ma- 














hogany 


3-77 


3.77 X 3.00 


■ 2,150 






32 


Mexican cedar . . 


3-75 


3.75 X 3.00 


1,950 






33 


Mexican mahogany 


3.75 


3-75 X 3.00 


4,500 






34 


White pine 


3.06 


6.20 X 4.75 


875 


" 




35 


White pine 


2.90 


4.73 X 4.00 


1,012 




Mean of two. 


^6 


Whitewood 


3-^5 


4 75 X 6 . 20 


900 




Mean of two. 


37 


Whitewood 


3 ^5 


4.75 X 4.00 


1,000 




Mean of four. 


^8 


Black walnut 


0.875 


4-75 X 4.00 


2,450 




Mean of two. 


39 


Black walnut 


0.87s 


4.00 X 3 94 


2,200 


'' 


Mean of two. 


40 


Black walnut 


0.875 


4 00 X 2.50 


2,525 




Mean of two. 


4t 
42 

43 


White oak 


2.40 
3-70 
3 90 


4.75 X 4.00 
4.75 X 4.00 
4.00 X 4.00 


3,550 

970 

1. 000 






Sprucfe 


'k 


Mean of four. 


Yellow pine 


tk 




44 


Black walnut 


0.75 


4 05 X 4 . 00 


2,8'>0 






45 


" " 


1. 00 


4.05 X 4.00 


2,56(4 






4b 




I '25 


4.05 X 4.00 


2,400 







4o8 



TIMBER IN COMPRESSION. 



[Art. 48. 



TABLE V. — Coniimied. 



47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 

59 
60 



KIND OF WOOD. 



Black Walnut. 



White pine 



Yellow birch . 
Yellow birch 
White maple. 
White maple 
White oak . . 



1.50 

1-75 
2.06 

0-75 
1.00 
1.25 
1.50 

1-75 
2.00 

4-25 
4-25 
4.00 
4.00 
3-95 



COMPRESSED 

SECTION IN 

INCHES. 



4 
4 
4 

4 

4 

4 

4 

4 

4 

4 

5 

3 

5-9« 

3-96 



X 4.00 
X 4,00 
X 4.00 
X 4.00 
X 4.00 
X 4.00 
X 4.00 
X 4.00 
X 4 00 
X 3.00 
X 3.00 
X 3.00 
X 3.00 
X 3.00 



ULT. RESIST,, 

LBS. PER SQ, 

INCH. 



2,SOO 
2,400 
2,360 
1,120 
1,100 
1.160 
1,070 

i,c6o 
1,000 
2,000 

1 /'SO 
1,700 
1,900 
2,500 



PERP. TO OR 
WITH GRAIN. 



Perp. 



Mean of two. 



The " ultimate resistance " was taken to be that pressure 
which caused an indentation of 0.05 inch. 

Nos. (44-55) show the effect of varying thickness of blocks. 
Within the limits of the experiments, the ultimate resistance is 
seen to decrease, somewhat, as the thickness increases. 

The results of the experiments given in this article show 
conclusively that the ultimate compressive resistance of short 
blocks of timber depend upon a number of conditions, such 
as method of compression, quality of material, size of block, 
etc., etc. These reasons account for the different results ob- 
tained by different experimenters for the same kind of timber. 



CHAPTER VII. 

Compression. — Long Columns. 

Art. 49. — Preliminary Matter. 

There is a class of members in structures which is subjected 
to compressive stress, and yet those members do not fail en- 
tirely by compression. The axes of these pieces coincide, as 
nearly as possible, with the line of action of the resultant of 
the external forces, yet their lengths are so great, compared 
with their lateral dimensions, that they deflect laterally, and 
failure finally takes place by combined compression and bend- 
ing. Such pieces are called *' long columns," and the applica- 
tion to them, of the common theory of flexure, has been made 
in Art. 25. 

Two different formulae have been established for use in 
estimating the resistance of long columns ; they are known as 
*' Gordon's Formula " and " Hodgkinson's Formula." Neither 
Gordon nor Hodgkinson, however, gave the original demon- 
stration of either formula. 

The form known as Gordon's formula was originally dem- 
onstrated and established by Thomas Tredgold (" Strength of 
Cast Iron and other Metals," etc.), for rectangular and round 
columns, while that known as Hodgkinson's formula (demon- 
strated in Art. 25) was first given by Euler. 

In 1840, however, Eaton Hodgkinson, F.R.S., published the 
results of some most valuable experiments made by himself, in 
cast and wrought iron columns (Experimental Researches on 
the Strength of Pillars of Cast Iron and other Materials; Phil. 



410 



LONG COLUMNS. 



[Art. 49. 



Trans, of the Royal Society, Part II., 1840), and from these 
experiments he determined empirical coefficients applicable to 
Euler's formula, on which account it has since been called 
Hodgkinson's formula. 

Mr. Lewis Gordon deduced from the same experiments 
some empirical coefficients for Tredgold's formula, since which 
time, Gordon's formula has been known. 

The latter is now in almost, if not quite, universal use 
among engineers, and will be completely given in the next Ar- 
ticle. Hodgkinson's coefficients and formula will be given 
farther on. 

Before taking up either, however, it will be useful and con- 
venient to determine the moments of inertia and squares of 
the radii of gyration of the various forms of cross sections of 
the columns now in common use. 

It will also be both convenient 
and important to determine the con- 
ditions which exist with an isotropic 
character of section in respect to 
the moment of inertia. 

In Fig. \a let BC be any figure 
whose area is A, and and whose 
centre of gravity is at O. In the 
plane of that figure let any arbitrary 
system of rectangular co-ordinates 
X' y Y' be chosen and let JCV be any other system having the 
same origin ; also, let ;r', y and ;r, / be the co-ordinates of the 
element D of the surface A, in the two systems. There will 
then result : 




Fig. l.a 



X ■= X cos a -\- y sin a, 

y := y cos a — x' sin a. 

The moments of inertia of the surface about the axes y and x 
will then be; 



Art. 49.] 



MOMENTS OF INERTIA. 



411 



\x^dA = cos^ a\x^dA -f- 2 sin a cos a x'y'dA + siii^ a y'^dA, 



y^dA = cof a y'^dA — 2 sin a cos ex x'y'dA -\- sin^ a 



x'^dA. 



If X and y are to be so chosen that they ^lvq principal 2iyiQS, 
then must \xydA = o, or : 



o 



xy dA = sin a cos a y'^dA + {(^os^ a — sin^ a) x'y'dA 



sin a cos a 



v'^dA 



{Id) 



tan 2 a = 



x'y'dA 



x'^dA — 



y\iA 



Hence, since /^;2 2a = tan (180 + 2<a'), there will always be 
two principal axes 90° apart. 

Now, if x'y dA = o, while no other condition is imposed, 

tan 201- = o. This makes or -- o or 90° ; i.e., X' Y' are the prin- 
cipal axes. 

If, however, x'y'dA = o, while ct is neither o nor 90°, Eq. 
(la) becomes : 



y'dA - 



x'^dA = o; 



or: 



412 MOMENTS OF INERTIA. [Art. 49. 

tan 2a = — , i.e., indeterminate, 
o 

This shows that any axis is 2, principal axis ; also, that : 
\x^dA ■=. \fdA = [x'^dA = l/'dA. 

Hence the surface is completely isotropic in reference to its 
moment of inertia ; or, ifs moment of inertia is the same about 
every axis lying in it and passing tlirougJi its centre of gravity. 

It has been seen that this condition exists where there are 
two different rectangular systems, for which 

\xydA = xydA = o ; 



but the first of these holds true if either x or y is an axis of 
symmetry, and the latter, if either x' or y' is an axis of sym- 
metry. 

Hence, if the surface has two axes of symmetry not at right 
angles to each other, its moment of inertia is the same about all 
axes passing tJirough its ce7itre of gravity and lying in it. 

Eqs. (\a) and the two preceding it also show that the same 
condition obtains, if the moments of inertia about fo2ir axes at 
right angles to each other, in pairs, are equal. 

In the case of such a surface, therefore, it will only be nec- 
essary to compute the moment of inertia about such an axis 
as will make the simplest operation. 

Since a column fails partly by flexure, it is manifest that the 
moment of inertia of its cross section should be the largest possible 
abotit an axis passing through its centre of gravity, and normal 
to the plane of flexure. 



Art. 49-] 



BOX COLUMN. 



413 



Box Column of Plates and Angles. 

Fig. I shows the cross section of "T 
a box column composed of 4 plates 
and 4 or 8 equal legged Ls. F B 

and CD intersect at the centre of ^"f lt~1 "J1 ^ 

gravity of the cross section. 

If the 4 Ls shown in dotted lines J 
are omitted, the moment of inertia 
about FB will be : 




I = -~ + bt 
6 



, {d + ij , (^ 4- /) d^ 

. -|- ^ 



'{s. — a) {d — 2df \- a {d — 2sY 



■ (I) 



If the dotted Ls are not omitted : 



j^bjj ^ ^^, {d + t'Y ^ {2s -V f) d^ 
62 6 



\s — a) {d — 2ay -V aid — 2sY 



■ ■ (2) 



If the 4 Ls shown in dotted lines are omitted, the moment 
of inertia about CB> will be : 



J- __ /'^3 a {w + 21 + 2sy (s — a) (zv + 2/4- 2ay 
^--6"^ 6 "^ 6 • 



(d — 2s) (w 4- 2/y _ dw^ . * 

12 U • • • • ^^^ 



414 



MOMENTS OF INERTIA. 



[Art. 49. 



If the dotted angles are not omitted : 

J. _ t'b^ a\{w -V 2t ^ 2sy — {w — 2sy\ 
^^^ + 6 



{s—d)\{w + 2t + 2df—{w—2df\ {d—2s) \{w + 2if—'uf\ . . 



12 



If latticing is used instead of the two plates bt\ f becomes 
equal to zero, and the first term in the second member of each 
of the above equations disappears. 

If A represents the area of the cross section, and r the 
radius of gyration: 



-i 



(5) 



L 1 1 ^ 




Box Column of Plates and Channels. 

Fig. 2 shows a normal cross sec- 
tion of this column. FB and CD 
— B intersect in the centre of gravity of 
, ^ the cross section. As in the pre- 
ceding Fig., these lines are lines of 
symmetry. The moment of inertia 
about FB is : 



I=^J1 + bt' (^ + ^'y -i- {s ■\- t)d^-s{d- 2ay ^ ^^^ 



The moment of inertia about CD is : 



J. _ fb^ 2a (w 4- 2/ 4- 2^)3 4- (^ — 2 a) {w + 2/)^ - dW^ , . 



Art. 49.] COLUMN OF PLATES AND ANGLES. 4^5 

If latticing takes the place of the two plates bt\ all terms 
in the second members of Eqs. (6) and (7) involving /' will dis- 
appear. The moment of inertia about FB then becomes : 

1 ^ ; . . . . \p) 

and that about CD : 

T — ^^ ^"^' + 2/ + 2^)3 + (d — 2a) {w + 2/)3 — dw"^ , ^ 

~" 12 

/ C 

{Radius of gyratioiif ~ r^ = — ; in which A 



A ' -f^ 



is the area of whole section. 

Eqs. (7) and (9) may also take the forms I 
given in Eqs. (15) and (16). j 



Built Colunm of Plates and A^igles. 



-1. 



F 



'r B 

< \ 

f 
I 






Fig. 3 shows a normal cross section of this k^ — t-^ — -^ 
column with the two axes of symmetry, FB Fig. 3. 

and CD, intersecting at its centre of gravity. 
The moment of inertia about FB is : 



/^^_^f£i + (rf+ ,.)=) + (-+ y^'-)^' 



2 \3 ^ V 6 

'{s — a) {d — 2ay> -{- a {d — 2sY 



. . . (10) 



The moment of inertia about CD takes the value : 



J. t'b-^ , a{2s ^- ty . (s - a) (2a + t)^ (d - 2s) t^ , . 
6 6 12 



4i6 



MOMENTS OF INERTIA. 



[Art. 49. 



If the two plates bt' are omitted, the terms involving t' in 
Eqs. (10) and (11) reduce to zero. 

{Radius of gy ratio rif = r^ := — ; in which A is area of sec- 

tion. 



-F- 



False Channel Section, 



I Let FB and CD intersect in the centre of 

r gravity, G, of the section. The distance x^^ of 
Jg. 4-B G from the back of the channel, is : 



f 



^^^_ y2\h^d-{b^-t^){d~2t')-\ ^ ^^^^ 



In which A is area of the cross section of 



Fig. 4. the channel. This is usually found by taking 

one-tenth of the weight, in pounds, per yard 
of the channel. Analytically : 



A = 2bt' + t {d — 2t') 



(13) 



The moment of inertia about CD then becomes : 



/' ^ 2/' {b - x,y + dx,^ -{d - 2t') {x, - t)^ . . 

3 

About FB^ it has the value : 

12 

{Radius of gyratiorif = r^ = — , 

The line CD can be very quickly and accurately located by 
balancing the section, cut out of manilla paper, on a knife 
edge. 



Art 49.] 



ANGLE IRON. 



417 



Eqs. (7) and (9) may now take the forms : 






■ (15) 



/= 2 



tv 



I' + A(- + .V, 



(16) 



In Eqs. (15) and (16) A represents the area of one channel 
section. • 



Angle Iron Section. 



J 



I 
\ 

F— J-- 



Fig. 5 represents this section 
with the two axes taken parallel 
to the legs, passing through the 
centre of gravity G. The area I 

of cross section is usually found ]._ 

from the weight per yard. Ana- 
lytically : 



K ? 



-E— J 



-■ i^— :pB 



-U-i 



(f \ I 

^ Fig. 5. 



A = It -\- {r - t)t = {i-\- 1' - t)t . . . (17) 

Again : 

_ y,[in ~ (I" - /') (/ - /)] 
x^ -^ . . . (isj 



^,^y^^„'-iV-t)iP-i^)-l _ _ ^^^^ 



The moment of inertia about CD is : 



27 



_ t {r - x,y + /^x^ - (/ - (^r - ^)' 



. . (20) 



41 8 MOMENTS OF INERTIA. [Art. 49. 

About FB : 

I ^ t Ji - ^J + i'^'' - {^' - 1) {x' - ty 

3 ' ' \ ) 

If the angle iron is equal legged, / becomes equal to /'. 
{Radius of gyratio7tf = r^ = — . 

As in the case of the C x^^ and x' may easily and ac- 
curately be found by balancing a model of the L section on a 
knife edge. 

Latticed Column of Four Angles. 

• 1 I ~^ The four Ls are held in the relative 

I positions shown in Fig. 6 by latticing, 

P l_g the latter being riveted to the legs of 

I the Ls, but not shown in the Fig. 
I The Ls are equal legged. 

From either Eq. (20) or (21), the 

•^ 7^ H moment of inertia of the section of any 

^' ' one L> about an axis passing through 

its centre of gravity and parallel to b, is : 

T _t{l - x:f + Ix,^ - (I - t) {x, - tj 

Hence the moment of inertia of the column section of Fig. 
6, about FB^ is : 

/■ = 4/. + ^ (^ _ x}j (22) 

A is the area of the column section, or four times the area 
of one L section. 

If b is different from b\ the moment of inertia of the col- 
umn section about an axis passing through its centre and par- 
allel to b' will be found by simply changing b' to b in Eq. (22). 



Art. 49.] 



LATTICED COLUMNS. 



419 



/' 



(Radius of gyrationf = r^ = —- , 



Latticed Columiis of Plates and Angles. 



c 



I' 1 

t i B 

I I 



.oL_. 



Fig. 7 represents a normal section 
j — ~] of one of these columns. By the aid of 
I Eq. (22), the moment of inertia of the 
section about FB may be written : 



L 5_.| J 

r D ' 

Fig. 7. 



/=/' + ^(^%(^' + /)^);. (23) 



and that about CD, remembering that 
in /', b' is to be changed to b : 



tb'^ 
/= /' 4- — • 



(24) 



If the plates are on the sides parallel to b\ then b is to be 
changed to b' and b' to /^ in Eqs. (23) and (24). 

Fig. 8 represents the normal section of the other of these 
columns, in which there is no latticing, the column being per- 
fectly closed. 

Again, using Eq. (22), the moment of ^^ 

inertia about the axis FB is : 



r 
I 



/=/'+! (^4 (^' + ^y 



F-4- 

i- 



+^:iK+i!i . . . (.5) -'- 



The moment of inertia about CD is : 



4- L.-B 



H — *-- i 4 

D 



Fig'. 8. 



/=/'+^-:(^)(g+(.+.r)+ 



tb^ 
~6 



(26) 



420 



MOMENTS OF INERTIA. 



[Art. 49. 



In the /' in Eq. 26, b' is to be changed to b. Ordinarily, 
b ^ b' and t = t' . 

(Radius of gyratiojif = r^ — ~ y A being area of cross- 
section. 



Tee Section. 

The axis FB is taken parallel to the head of the tee section, 

and CD perpendicular to it, 

\ 5 !? ^ while G is its centre of gravity. 

Analytically, the area of the 
section is : 



-T' — 



4=prx:-: 



A = bt -^ dt' . . . (27) 



D 

Fig.9 



1 The area may also be taken 

from the weight in the usual 



X, = 



manner. 

_^2[{b-t'y^-{d-^tyt'-] 



A 



. . . (28) 



The moment of inertia about FB is : 

^ ^ bx,^ + t'{d + t - x:f -(b- t') {x, - ty 

3 
The moment of inertia about CD is : 

tb^ + dt'^ 



1 = 



(29) 



12 



(30) 



{Radius of gyrationf = r^ z= — . 

As in the other cases, FB may be located by balancing on a 
knife edge. 



Art. 49.] 



FALSE EYE SECTION. 



421 



False Eye Section. 

If the area is not taken from the weight per yard, it may be 

written : 

• (c 



A=bd-{b- t') {d - 2t) , . (31) 
The moment of inertia about CD is : 



r &t ■»! 



^_ 2/^3 -^ [d — 2t)0 
■'■ • • • 

12 



(32) 



About FB it has the value : 



J ^ bd^ - {b - f) (d - 2ty' 
12 



I « 






D 

Fig.lO 



. • . (33) 



{Radius of gyratiorif = r^ = — , 



Star Section. 

Fig. II shows this section with the different dimensions. 

The area of cross section is : 



I 
I >1 



I 1 



F- + -f 



-i— 



A = bt -\- b't' - tt' . . . (34) 



-5 *i 



I The moment of inertia about 

FB is : 



-B 



Fig.n 



/ = ''" + !t - '^' . . . (35) 

About CD the moment of iner- 
tia has the value : 



422 



MOMENTS OF INERTIA. 



[Art. 49. 



~ 12 



(36) 



Ordinarily, ^ = /'. 



/ 



(Radius of gyratiorif = r^ = — . 



Solid Rectangular Section. 




In Fig. 12 ^ = M. 
The moment of inertia about 
FB\s\ 

/-Y^; . . (37) 

and about CD : 



/ = 



12 



(38) 



/ /i^ b^ 
{Radius of gy ratio nf = r'^ =z — = — or — 

jrL 12 12 



If the rectangular section is square, b = h. 



Hollow Rectangular Sections, 



The area of the section shown in Fig. 13 is: ^4 r= bh — b'/t. 
The moment of inertia 
about FB is : 



bk3 _ ^7/3 

^= — Y-^ — ; . . (39) ^ 





c 








1 

1 




1 










h 
1 

-hB 

1 
1 


' 


1 

1 




\ 



and that about CD is : 



^ .__^- 

Fig. 13. 






Art. 49-]' 



CIRCULAR SECTIONS. 



423 



/ = 



hb^ - hb'^ 



12 



(40) 



/ 



3 



K-f J 



(Radius of gyratioiif = r^ = 

All the equations of this case (except Eq. (40)), just as they 
stand, apply directly to the rect- 
angular cellular section of Fig. 14, 
considered in reference to the axis 
FB. If there were n cells instead 
of 3, the space between any adja- 
cent two would have the width 

n ' 



3 



Fig.H 



Solid and Hollow Circular Sections. 

First consider a solid cylindrical column whose cross section 

has the radius r^^ as shown in Fig. 15. 
The moment of inertia about any di- 
ameter is : 

7rr/ 




/ = 



{Radius of gyrationf = 



(41) 



Ttr^'* 
47rr, 



Fig. 15. 4 

Next consider a hollow circular column whose interior and 
exterior radii are r^ and r^ respectively. The moment of iner- 
tia about any diameter is: 

/ = -^-^— '-^ = ^ ' J '- ' ; {A = area) . . (42) 



(Radius of gyratiorif = 



I 

7r(r/ - r,') 



+ r. 



= r\ 



424 



MOMENTS OF INERTIA. 



[Art. 49. 



As tables of circular areas are very accessible, it may be 
convenient to write : 



7^ = 



^^2^ . or r^ =r ^(^ / -f- ^i') 
12.566' 12.566 



Phcenix Section. 

Fig. 16 shows the section of a 4 segment Phoenix column. 

Let CD represent any axis 
taken through the centre of the 
column. The moments of iner- 
tia of the rectangles bl about 
axes through their centres of 
gravity and parallel to CD will 
be very small indeed compared 
with the moment of inertia of 
the whole section. The mo- 
ment of inertia of any one 
of these rectangles, therefore, 
about CD^ will be taken as 
equal to the product of its area 
by the square of the normal 
distance from its centre of gravity to the axis CD, The mo- 
ment of inertia of the section about CD will then be : 




4 



r,-\ j sin^ " + ( ^2 H I '^os' a 



••• ^ = ^ + ^^^ V' + 2. 



(43) 



The moment of inertia is thus seen to be the same about 
all axes, a result of the general principle established in the first 
part of this Article. 

The area of the cross section is : 



Art. 49.] 



TRUE EYE SECTION. 



425 



A = n{r^ - r/) + ^bl. 



(43^) 



{Radius of gyratiorif = r' = -^ . 



The moments of inertia of six and eight segment columns 
may be found in precisely the same manner. The moments of 
inertia of the rectangular sections of the flanges about axes 
passing through their centres of gravity, being very small indeed 
when compared with the moment of inertia of the whole sec- 
tion, may be neglected without sensible error. 



Let r — 



2S 



True Eye Section, 
; r is then the 



b ~ t^ 
batter, or slope, of the under side 
of each flange to the top or bot- 
tom of the beam ; it ranges from 
about one-third to essentially noth- 
ing. 

If the area of the cross section 
is not deduced from the weight : 

Area of section 

= A — 2bt '\- ht^ + s{b — /,) . (44) 

The moment of inertia about ' 




Fig.17 



__ 2/^3 ■!■ k^t,^ r{b^ - 1/) , 

^ - - + ~^8 • • . • (45; 



12 



If t^ is very small as compared with b, rem^embering that 
r is then essentially equal to s^ there will result : 



426 



MOMENTS OF INERTIA. 



[Art. 49. 



12 



(46) 



This formula is sufficiently accurate for all wrought-iron 
and steel beams. 

The amount of inertia about FB is : 



/ 



4r ^ ^ 



12 



In any of these three cases : 

/ 



{Radius of gyrationf = 



A 



(47) 



(48) 




True Channel Section, 
In Fig. 18 let r : 



; as before, 



I - b - /, 

— ^ 1 it is the batter or slope of the under side 

! ! of the flange. ' 

1' I If the area of the section is not de- 

I 



^ j i — B duced from the weight : 




Area of section 
= A = 2bt -\- ht, + j(^ -/,)... . (49) 



The centre of gravity, G, can be found 
by balancing a manilla, or other, pattern 
Fig.18 on a knife edge ; or, analytically : 

^^^ ^ b'i + y^ K' -^-'AKb ~ o {b + 2t,) ^ ^ ^^^^ 

A 



The moment of inertia about CD is : 



Art. 49.] TRUE CHANNEL SECTION. 427 



If /j is very small compared with ^, and remembering that 
br is then essentially equal to s ; this last equation will become : 

i='^li±^iJ)t±Jb^_A.,^. . . . (52) 

The moment of inertia about FB is : 



/ = ?^^ (53) 

12 

In any of these three cases : 

{Radius of gyratioiif = ~j (54) 



Deck Section. 

The head of this section will be considered circular in out- 
line, as shown in Fig. 19. Let a be the area of the circle C. 

If the area of the section is not deduced from the weight : 
Area of section 

= A=a-\-{d- h)t, ^ {b - t:) {t -{- y2 s) , . (55) 

If the centre of gravity, Gy is not found by balancing a 
pattern on a knife edge, there will result, analytically : 

- _ ^{2d - Ji) -\-t,{d- Jif + bt- + s{b - /,) it^y^s) ... 



428 



MOMENTS OF INERTIA. 



[Art. 49. 



.D-j 4g— 




.IF 



^-H 




-t-. h 



! C 



tfs>\ I Fig,i9 



The moment of inertia about FB is : 



^=^4;+('^-3V'-^^' 






(57) 



in which equation r = 



2S 



The moment of inertia aboGt CD is : 



}( aJe -^ t^ {d - h - t - s) -^ th^ -\- - {b^ - tf) 
1= .(58) 



12 



br 
If /- is small as compared with b, so that essentially — = s : 



J ^ ZctJf + Atj' {d - h ~ t -s) -^ (4/ + s)b^ ^ , . 

48 



In all cases : 



Art. 49- J ANGLE SECTioisr. 429 

(Radius of gyratiorif = — (60) 



Angle Section about Oblique Axis, 

The angle iron is here supposed to be equal legged, and the 
axis about which the moment of inertia is taken, passes through 
the centre of gravity (before found in this Art.) and cuts the 
sides / at an angle of 45°. In Fig. 20, G is the centre of grav- 
ity and HK the axis. 




The moment of inertia about HK is : 

jr^ 2l^/ _ (^^ _ ty\ + /{/ - (2X, - y^ t)\ ^ ^ ^^^^ 

If A is the area of cross section : 

(Radius of gyratioTif = ^ (62) 

If a long column has the same degree of fixedness or free- 
dom in all directions, the least value of the square of the ra- 
dius of gyration must be taken for insertion in Gordon's for- 
mula; because in the plane of that radius the column will offer 
the least resistance to flexure. 



430 GORDON'S FORMULA. [Art. 50. 



Art. 50. — Gordon's Formula for Long Columns. 

Since flexure takes place, if a long column is subjected to a 
thrust in the direction of its length, the greatest intensity of 
stress in a normal section of the column may be considered as 
composed of two parts. In fact, the condition of stress in any 
normal section of a long column is that of a uniformly varying 
system composed of a uniform stress and a stress couple. In 
order to determine these two parts let 5 represent the area of 
the normal cross section ; /, its moment of inertia about an 
axis normal to the plane in which flexure takes place ; r, its 
radius of gyration in reference to the same axis ; P, the magni- 
tude of the imposed thrust ; f, the greatest intensity of stress 
allowable in the column, and A, the deflection corresponding 
io f. 'Let p' be that part of /"caused by the direct effect of P, 
and/" that part due to flexure alone. Then, if k is the greatest 
normal distance of any element of the column from the axis 
about which the moment of inertia is taken, by the " common 
theory of flexure :" 

cPA=^; ., p ^_^_ .... (I) 

If the column ends are round, c' = i ; but if the ends are 
fixed, the value of c' will depend upon the degree of fixedness. 
Also, ^ 

Hence, 

P= l^ (3) 

1 + -^ 



Art. 50.] 



GORDON'S FORMULA. 



431 



Eq. (3) may be considered one form of Gordon's formula. 
Before deducing the more common and useful form, it will 

be necessary to show that A = « -y ; in which expression a is 

considered constant. 

Let / be the greatest intensity of bending stress in any 
section, whose greatest value in the column is /". By the 
" common theory " (taking the origin of co-ordinates at the 
centre of gravity of the cross section at one end of the column, 
and the axis of x along the centre line before flexure) : 



EIp-=fl. 
dx^ It 



Also, 



^ Mh . ., MJi 

/ = -^ , and / ^ ~j~ 



(4) 



in which equations E is the coefificient of elasticity M the 
bending moment for any section, and M^ the value of M cor- 
responding to /". 



Hence, 



/=/ 






an 



dx' Ek M^ ' 



Consequently, 



i.hh M. 



■ (5) 



The section located by 4 is that at which the deflection is 

greatest, and for which -f- = o, while —= is considered con- 

ax Ell 

M . 
stant. The ratio -^ is numerical, though variable, being one be- 



432 GORDON'S FORMULA. [Art. 50. 

tween quantities of the same degree. M^ is exactly the same 
as M^ except that x^ in the latter, is displaced by 4 ; there are 
the same number of terms in each, and those terms are multi- 

plied by the same coefficients. Now M dx^ may be so ar- 

ranged as to have the same number of terms as M^^ but the co- 
efficients of tJiose terms will be different, and the exponents of l^ 
in the former will be greater by 2 tJian the exponents of 4 ifT' M^, 
Hence // = c^l^ {c being some constant) will be a factor in all 
the terms of the definite double integral. From these con- 
siderations it follows that, 



Mdx^ 



in which d is some constant. Consequently, 



dl^ ; (6) 



^ = -ir^ ='^-T •••,••• (7) 



It is seen therefore that the quantity a^ depends upon both 
/" and E, and it is ordinarily considered constant. 
Since / = 5r^ Eqs. (i) and (7) give : 

c'Pl^ P l^ P f P\ 



Eq. (8) shows that a^d = a. 
Hence, 

^=-^ (9) 

I ■\- a — 



Art. 50.] 



ROUND ENDS. 



433 




The integration by which Eq. (7) is obtained, being taken 
between limits, causes everything to disappear which 
depends upon the condition of the ends of the col- 
umn. Consequently Eq. (9) applies to all columns, 
whether the ends are rounded or fixed. Let the lat- 
ter condition be assumed, and let it be represented in 
the adjoining figure. Since the column must be bent 
symmetrically, there must be at least two points of 
contraflexure. Two such points, only, may be sup- 
posed, since such a supposition makes the distance 
between any two adjacent points the greatest possible 
and induces the most unfavorable condition of bend- 
ing for the column. 

If B and C are the points of contraflexure sup- 
posed, then BC^WX be equal to a half of AD, for each 
half of BC must be in the same condition, so far as flexure is 
concerned, as either AB or CD. Also, the bending moment at 
the section midway between B and C must be equal to that at 
A or D. Consequently, the free or round end column BC 
must possess the same resistance as the fixed or flat end col- 
umn AD, In Eq. (9), therefore, let / = 2BC = 2/^ : 




Fig.l 



P = 



_ /^ 






(10) 



Eq. (10) is, consequently, the formula for free or round end 
columns with length /j. 

The flat, or fixed end column AD, is also of the same re- 
sistance as the column AC, with one end flat and one end free 
or round. Hence in Eq. (9) let there be put / = \ AC ~ |/', 
and there will result, nearly, 



P 



fS 



I + 1.8^ — 
^2 



(II) 



28 



434 GORDON'S FORMULA. [Art. 50. 

Eq. (11) is, then, the formula for a column with one end 
flat and the other round. A slight element of approximation 
will ordinarily enter Eq. (11) on account of the fact that C \s> 
not found in the tangent at A just as Eqs. (9) and (10) are 
based on the supposition that A and D lie exactly in the line 
of action of the imposed load. 

If the column is swelled, as shown in Fig. 2, the 
the moment of inertia /, and distance, //, become vari. 
able. Hence : 

P = ~j~ y and / =: -p . 



Consequently, 



and, 



^ ^ AUu I 



rL 



A = 



^^-ydx^. . . . (12) 



Fig.2 

M 
If, in the reasoning applied to Eq. (5), there be written — 

M 
for M^ and -y? for M^, it will at once be seen that Eq. (12) will 

give precisely the same general form of result as Eq. (5), but 
the coefficient a will have a different value. Farther, since 
lo -^ I can never be less than unity, but is in general greater, it 
follows that, for swelled columns, a is greater than for columns 
that are not swelled. Although these considerations show that 
the value of a w411 be different in the two classes of columns, 
yet they also show that the general form for the breaking 
weight P, whatever may be the condition of the ends, will be 
precisely the same whether the columns are swelled or straight. 
Since the swelling of a column will give it a greater resist- 
ance to bending,/" will take a correspondingly less value, while 



Art. 50,] VARIABILITY OF ''CONSTANTSA 435 

Pand 5 remain the same. Eq. (8), then, shows that if /"and S 
are unchanged, P must be increased. In other words, a swelled 
column will sustain a greater load than one not swelled but 
possessing the same kind and area of cross section. This is 
indeed true of solid columns, but may not be, and usually is 
not, for reasons to be assigned hereafter, true for built columns 
of shape iron. These reasons are not introduced in the hy- 
pothesis on which the formulae are based. 

Although the quantities /"and a^ in Eqs. (9), (10) and (11), 
are usually considered constant, they are strictly variable. Eq. 
(7) shows that <^ is a function of /'' -^ E. It is by no means 
certain that/" is the same for different forms of cross section, 
or even for different sections of the same form, and the very 
variable character of the coefficient of elasticity is well known. 
It (the latter) is known not only to vary with the products of 
different iron mills, but even with the different products of the 
same mill. 

Again, the greatest intensity of stress, /", which can exist in 
the column varies not only with different grades of material, 
but there is some reason to believe that it must also be consid- 
ered as varying with the length of the column. The law gov- 
erning this last kind of variation, for many sections, still needs 
empirical determination. It is clear, therefore, that both/" and 
a must be considered empirical variables. 

The expense necessarily attending experimental researches 
on the ultimate resistance of long columns built of American 
material, has prevented the attainment of many desirable re- 
sults. Yet much very valuable work of this kind has been 
done. 

In the ^' Report of Progress of Work," etc., made by Thomas 
D. Lovett, consulting and principal engineer to the trustees of 
the Cincinnati Southern Railway, Nov. i, 1875, are found the 
records of some valuable experiments on wrought-iron long col- 
umns. The results of these experiments will be used in fixing 
values of a and/". 



43^ GORDON'S FORMULA. [Art. 50. 

If the number of experiments were sufficiently great, the 
results should be combined by the " Method of Least Squares." 
In the present instance, however, the use of the method is al- 
together impracticable in consequence of the small number of 
experiments of any given class. It will be seen, however, that 
the combination of the experimental results is not altogether of 
a random nature. 

Since /and a are to be considered variable quantities, let j^ 

p 
take the place of /"and x that of a ; also, let p = — represent 

the mean intensity of stress. Eq. (9) then takes the form : 



^ = r-fcr' ('3) 



in which c =^ l^ -r- r^ For round or free end columns x will 
take the place of 4^, and of i.8« for columns with one end round 
and one end flat. 

In Eq. (13) there are two unknown quantities, j^ and x, con- 
sequently two equations are required for their determination. 
If two columns of different ultimate resistances per unit of 
section, and with different values of c, are broken in a testing 
machine, and the two sets of data thus established separately 
inserted in Eq. (13), two equations will result which will be suf- 
ficient to completely give j/ and x. Those two equations may 
be written as follows : 

y = p' (I -^c' x) (14) 

y^fii^c'x) ...... (15) 

The simple elimination of j/ gives: 

c p — c p 



Art. 50.] PIN ENDS. 437 

Either Eq. (14) or (15) will then give j. 

In selecting experimental results for insertion in Eq. (16), 
care should be taken to make the differences/" — /' and c — c" 
as large numerically as possible, in order that the errors of ex- 
periment may form the smallest possible proportion of the 
first. 

Before applying Eq. (16) it would be well to recognize the 
condition of the end of a column resting on a pin, as in pin 
connection trusses. The end of a column resting on a pin 
might, at first sight, be considered round or free in a plane 
normal to the axis of the pin. The compressive strains exist- 
ing in the vicinity of the surface of contact between the pin 
and soffit of the pin hole, produce a considerable surface on 
which the frictional resistance to any relative movement is very 
great. This resistance to movement is not sufficient to pro- 
duce a " flat " or " fixed " condition of the column end, but 
causes a degree of constraint intermediate between the flat and 
round condition; so that a column with two " pin ends " gives 
an ultimate resistance approximating to that of a column with 
one round end and one fixed end. The following two cases 
will then hereafter be recognized : 

Two Pin Ends, 

One Pin End and one Flat End. 

All the necessary data for the treatment of the experiments 
given in the report of Mr. Lovett, are found in the following 
table. The column " Area " gives the areas of normal cross sec- 
tions in square inches. The column r* gives the squares of the 
radii of gyration, in inches, about axes normal to the plane of 
bending. It is inferred from the table and the report under 
consideration that the radii of gyration for swelled columns 
belong to sections at middle of columns. The c column con- 
tains the squares of / divided by r, both being taken in the 
same unit ; it is a matter of indifference what that unit may be. 



438 



GORDON'S FORMULA. 



[Art. 50. 



The quantities Xy y and/ -d^x^ found by the fonnulcB (16), (15) or 
(14) and (13). The column headed Exp. contains the ultimate 
resistances in pounds per square inch, determined by experiment. 




Keystone 



Cloned 




Open 



Square 



"^ 



^ 



£L 



l*hoen 130 A.m.Br. Co. 



An ** open " column is one in which the flanges of the seg- 
ments that compose it are separated by an open space; a 
closed column is one in which no such spaces are found. All 
the columns treated in the table are closed except Nos. 2, 3, 4, 
8, 25,31, 24, 26, 30, and 5. 

The columns 13 and 19 failed about axes giving th.& greatest 
moments of inertia or radii of gyration. This was probably 
due to some cause equivalent to a less degree of constraint at 
the ends than was intended. For this reason those two results 
are not used in determining x andjK. They. will be noticed 
again. 

An examination of the table shows that the flat end swelled 
and open straight Keystone columns give about the same ulti- 
mate resistance, by experiment, per square inch, so long as c 
remains the same, though' the straight columns give the largest 
results by a little. Hence /' was taken as an arithmetical 
mean of the experimental results of Nos. 4, 25, 31, 24, 26, and 
30, and c' at 9,208. In the same manner/" was taken a mean 
of the experimental results for Nos. 3 and 8, and c" at 3,060. 
The arithmetical means mentioned are/' = 25,517 pounds, and 
/" = 32,850 pounds. Substitutions in Eqs. (16) and (15) or 
(14), then give : 



X = 0.00005455 andj^ == 38,300.00 pounds. 



Art. 50.] 



EXPERIMENTAL RESULTS. 



439 



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440 GORDON'S FORMULA. [Art. 50. 

y is taken at the nearest hundred. These values of x and /, 
placed in Eq. (13), give the values in column ''/." 

Since, however, the resulting values of ''/ " were a little too 
large for the swelled columns, and a little too small for the 
straight open ones, x was allowed to remain as determined, 
and y was made 36,000.00 for swelled, and 39,500.00 for 
straight open columns. The resulting values of ^^ p " are given 
in the table. 

The differences between the results in the columns **/" 
and '■^ Expy are not greater than experimental differences. 

Since x depends on the condition of the ends of the col- 
umns, as well as on the character of the iron, it is reasonable to 
give it the same value for all flat end Keystone columns. Then 
taking jj/ at 39,500.00 pounds, it will be seen that Eq. (13) gives 
results agreeing, as nearly as could be expected, with those of 
experiment for straight closed Keystone columns with flat 
ends. 

No. 5 is the only experiment with a pin end Keystone col- 
umn. As it was also swelled, y has been taken at 36,000.00 
and X at T-jioi7, so that / would be a little less than the result 
of experiment. As these values depend on one pin end ex- 
periment only, they should not be considered very satisfactory. 
At the same time corresponding values for other columns show 
that they cannot be very erroneous. 

Precisely the same general principles and considerations 
governed the selection of x and y for the several remaining 
classes of columns shown in the table. The agreement be- 
tween the columns/ and Exp. is as close as could be expected. 

The extraordinary character of Nos. 13 and 19 has already 
been noticed. No. 13 was intended to be a pin end column, 
but the plane of flexure contained the axis of the pin. Now 
if it be considered a round end column in the plane of failure, 
X will have the value 4 X T-euoTT = tttot^ ^^^ the resulting 
value of p will be 24,600.00 pounds. The result of experi- 
ment was 24,000.00 pounds. Again, No. 19 was intended to 



Art. 50.] RESULTING FORMULA. 44I 

be a flat end column, but it failed in the direction of its great- 
est radius of gyration. Using the values of x and y for pin 
ends, there will result / — 26,400.00 pounds. The result of 
experiment was 27,800.00. The effect of defective fitting, etc., 
would therefore seem to be the lessening of the end constraint 
by what may be termed one degree. 

Expressing all the results in concise formulae, they may be 
written : 

Keystone Columns, 
Flat Ends— Swelled / = '^^^ ^, ; ... (17) 



18300 r 



Flat Ends— j Open. ) ^ _ 395QQ . /.^ 

Straight.. (Closed ( ^ " i /^ ' * * * ^^^^ 

I _| . 

18300 r* 



Pin Ends — Swelled -P '=■ "Tz ' • • • (^9) 

15000 r* 



Square Columns. 

Flat Ends p = '^^^^ ^, ; . . . (20) 

35000 r^ 



Pin Ends ^^__i9000__. ^ ^ ^^^^ 

I + 



19* 



17000 7* 



442 GORDON'S FORMULA. [Art. 50. 



PJicenix Columns. 

Flat Ends / = — ; . . . (22) 

I +— ^ — 
50000 r'^ 

Round Ends / = j^ ; . . . (23) 

I H — 

12500 r* 

Pin Ends (hypothetical) . . . .p = ~ ; . . . (24) 

I +_l_i! 
22700 r"" 



American Bridge Co. Columns. 

T-1 . T- 1 . 36000 , , 

Flat Ends p — ; . . . (25) 

46000 r^ 

Round Ends p ■= ^ r ; . . . (26) 

, I /^ \ / 

I + ■ ^ 

1 1 500 r^ 

-n- T- 1 i. 36000 , . 

Pin Ends / = ^ — _ -; . . . (2;) 

I + • 

21500 r^ 

The pin end formula for the Phoenix column is based on 
the hypothesis that the relation between the values of x for 
flat and pin ends is the same as that existing in the American 
Bridge Co. columns, which last is shown by experiment. This 
is a very unsatisfactory method, and should not be implicitly 
relied upon. 



Art. 50.] SWELLED COLUMNS. 443 

All values of x for round end columns are found by multi- 
plying the corresponding flat end quantities by 4, according to 
Eq. (10). 

Eqs. (17) to (28), inclusive, give the ultimate resistances of 
the various classes of columns. With great variations of stress 
a safety factor as high as six or eight may be used, or it may 
be as low as three or four if the condition of stress is uniform 
or essentially so. 

For a complete account of the details of the foregoing ex^ 
periments, the original " Report " must be consulted. The 
consideration of the shades of influence exerted by the differ- 
ent devices to produce a given end condition have here been 
neglected on the ground that such degrees of influence are too 
small to be involved in a practical formula. 

Some important deductions bearing on built columns of all 
forms of cross section may be drawn from the results of these 
experiments. It has already been noticed that the swelled 
columns Nos. 2, 3, 4, 8, 25, 31, do not give as great ultimate re- 
sistances as similar straight ones; a result perhaps not to be 
expected, though the explanation is simple. Both internal 
tensile and compressive stresses are induced in the originally 
straight segments w^hen they are sprung to their proper curva- 
ture in the swelled column. Consequently this internal com- 
pressive stress causes a portion of the material to reach its 
ultimate resistance much sooner than would be the case if the 
columns were straight. Again, a slight increase of direct com- 
pressive stress is caused by the inclination of the segments of 
the column to its axis. If the segments could be prepared for 
the column without initial internal stress, the ultimate resist- 
ance would probably be considerably increased. 

A consideration of these experiments would also seem to 
indicate that a closed column is somewhat stronger than an 
open one. This is undoubtedly due to the fact that the edges 
of the segments are mutually supporting if they are brought in 
contact and held so by complete closure, but not otherwise. 



444 GORDON'S FORMULA. [Art. 50. 

Thus the crlpphng or buckling of the individual parts of the 
column is delayed, and the ultimate resistance increased. 

The general principles which govern the resista?ice of built 
columns may, then, be summed up as follows: 

The material sJwiild be disposed as far as possible from the 
neutral axis of the cross section^ thereby increasing r ; 

There shotdd be no initial internal stress ; 

The individual portions of the column should be mutually sup- 
porting ; 

The individual portions of the column should be so firmly 
secured to each other that no relative motion can take place, in 
order that the column may fail as a whole, thus maintaining the 
original value of r. 

These considerations, it is to be borne in mind, affect the 
resistance of the column only ; it may be advisable to sacrifice 
some elements of resistance in order to attain accessibility to 
the interior of the compression member, for the purpose of 
painting. This point may be a very important one, and should 
never be neglected in designing compression members. It 
may be observed, however, that the sole object is to prevent 
oxidation in the interior of the column, and if the column is 
perfectly closed this object is attained. Phoenix columns 
which have been in the most exposed situations (in one case 
submerged in water at one time for several hours) during 
periods varying from twelve to twenty years, without the 
slightest oxidation in the interior of the columns, have come 
within the observation of the writer. Different results, how- 
ever, in other cases have been found. 

In the experiments detailed in Mr. Lovett's report it is to 
be noticed that all deduced values of j^ are less than the ulti- 
mate resistance of wrought iron in short blocks, and some, 
though not nearly all, would seem to indicate that this differ- 
ence increased slightly with the length of the column. Fur- 
ther experiments, therefore, may show that the quantity /"has 
some such value as the following: 



Art. 50.] BOUSCAREN'S EXPERIMENTS. 445 

(7 being a constant quantity, and /"a function of the reciprocal 
of the length. 

In connection with the experiments already detailed, Mr. 
G. Bouscaren, C.E., has given an account, in the Trans, of the 
Am. Soc. of Civ. Engs. for Dec, 1880, of other experiments, 
the results of v/hich are given in the table below. 

Column No. 33 was composed of four angle irons, 

arranged as shown in the figure. It was swelled —^jj-p^- — "^^ 
from %y." X 83^" at the ends to 10" X 10" at \^ \ 

the centre. There was only one experiment ^''} [ 

with this form of column, consequently the val- I^f^ JP 
ues of X and y in Eq. (13) could not be deter- ~ t---"io''~~-^ 
mined. The angle irons, however, were of the 
same manufacture as the iron of which the Am. Br. Co.'s col- 
umns were built. As a mere matter of trial, therefore, y is 

taken at ^6,000.00 pounds, and x is then found to be . 

^ ^ 43000 

This result seems to indicate considerable advantage in such 

a form of column, but one experiment alone furnishes insuf- 

cient basis for such a deduction. 

The columns 35 and 36 illustrate the effect of repeated stress. 

The columns 37 to 43, inclusive, were intended to furnish 
information in regard to the distance between the rivets in the 
zigzag bracing and the thickness of the metal, in order that 
the column may fail as a whole and not by '' local buckling." 

Columns 39 and 40 were each composed of a single short 
piece of channel bar ; the others were composed of two chan- 
nel bars held together by zigzag bracing. 



446 



GORDON'S FORMULA. 



[Art. 50. 



NO. 


LENGTH. 


AREA. 


r2. 


tr. 


X. 


:>'• 


> 


Exp. 




33 
35 
36 
43 
37 
38 
39 
40 

41 
42 


28' 6" 
34' 0" 
34' 0" 
26' 7" 

27' 6" 

23' 00'' 

27' 6" 
27' 6" 


r:3" 
5.68 
7.48 
7.48 
6.50 
12.08 
13 48 
6.6 
6.6 

13-74 
11.05 


20 

8 

8 

5 

'9 

?o 





20 

21 


^7 

73 

73 

95 

98 

69 

7 

7 

7Q 

i6 


5.828 


I 
431100 


36,000 


31,700 


3^7"o 
20,053 
23,128 
18,000 
2gs6oo 
32,300 
35,400 
35,703 
32 , 400 
32,300 


Pin Ends, 

Flat Ends. 

It 11 






The following forms of cross section, and observations, are 

taken from Mr. Bouscaren's account : 

^0' 35* — Gave way by pin crushing and splitting 

web of channel. Column not injured otherwise. 

No. 36. — Column No. 35 tested again after 

crushed ends had been cut off and thickening plates 

riveted on with pin holes 34 feet from centre to 

' centre. Column failed by deflection. 

J Fit 3f* 

N'o. 43. — Failed by bending sideways at right j i ,1 

angles to pins, without buckling of metal. Bracing ^i^\ \ 



-^ ^ ■a. 



13, 



X 



/^" ; rivets 20" apart in same flange and 10'' j 1 ^a'' 
in opposite flanges. 



li) 



^■^ 
I 



<fn?s 



1 1 



^^' 37- — Webs buckled in both directions, 

-r-^i in middle and one end of column ; column did 

j^ 12- |--^ ^Q^ bend. Bracing 2" x y^" ; rivets 24" apart 



I [^■■■■-■■)<_ '■III ■!! 



i^l in same flange and 12" in op- 
posite flanges. 
A^o. 38. — Failed in same manner as No. 
37 and by deflection, simultaneously. Bracing 
and rivets same as in No. 37. 



1 Y 12 -jH 

I ■• I o V 1 

¥1 .,,, ? 



_>k^^ 



■^ 



^1 



L 



Hf- 



_.,^^ No. 39. — Failed by buckling of web and 

w '^°'' J .^ir flanges. 

Failed by 



12 ^ 

I 



No. 



40. — Same as No. 39. 
buckling of web and flanges. 



Art. 50.] BOUSCAREN'S CONCLUSIONS. 447 



N'o. 41. — Column same as No. 37 with riv- -r^t dr> 

ets spaced 20", in same flange, instead of 24", i ,j^ 12':- \J^ 



Failed by buckling of web and bending in } [ ^0:^35 i 1 
both directions, simultaneously. —^-^ j'j 

__^ No. 42. — Failed by buckling in plane of lat- 

I !' z — T~^P ticinff, without buckling of metal. 

.{i\ 8' I From these experiments Mr. Bouscaren 

{ ! | . x— L_! concluded that, for the ratio of length to diam- 
eter used, " the thickness of metal should not 

be less than — of the distance between supports transversely, 
30 

. . . . and that the distance between rivets longitudinally 
should be such that the length of channel spanning it, con- 
sidered as a column, .... shall give the same resistance 
per square inch of area as the column itself, treated in the 
same manner with the same constant ^y*",'' {y)- 

These conclusions are agreeable to that reached by Mr. B. 
B. Stoney : '' When the length of a rectangular wrought-iron 
tubular column does not exceed 30 times its least breadth, it 
fails by the bulging or buckling of a short portion of the plates, 
not by the flexure of the pillar as a whole." (Theory of Strains, 
2d Edit., Art. 334.) 

It should be stated that the experiments whose results have 
been given were made in hydraulic machines in which the 
forces were not weighed, consequently the results involve the 
"packing" friction, which was probably not great, however. 

In applying Eqs. (9), (10), and (ii) to solid cast-iron col- 
umns, there may be taken, approximately: 

f = 80000.00 pounds, and a = -qzito' 
For solid wrought-iron columns, approximately: 
/ = 36000.00 pounds, and a = ^ji'^'o • 



448 



GORDONS FORMULA, 



[Art. 50. 



Experiments on steel columns are still lacking. Mr. B. 
Baker, in his *' Beams, Columns, and Arches," gives for 

Mild Steel, / = 67000.00 pounds, and a = -g-gToT* 
Strong Steel,/ = 114000.00 pounds, and a = yjJoo-- 

These, however, must be considered only loose approxima- 
tions for the ultimate resistance. 

In the ** Trans, of Am. Soc. Civ. Engrs.," for Oct. 1880, are 
given the following formulae for ultimate resistance of wrought- 
iron columns, designed several years since by C. Shaler Smith, 
C.E.: 

• 

Square Cohimn, 



/ = 



FLAT ENDS. 
38500 



I + 



5820 d^ 



/ = 



ONE PIN END. 
38500 



I + 



I I' 

3000 d^ 



P = 



TWO PIN ENDS. 

375Q O 
I 



1900 d"^ 



I +—-3-, 



Phcenix Column, 



42500 



I + ----7^ 



4500 d'' 



P 



40000 



I 4- • - 

2250 d"" 



^6600 



I + 



I'' 



1500 d"^ 



American Br, Co, Column, 



/ = 



36500 



I + 



I I' 
3750 d' 



P 



3^500 

I A - 

2250 d^ 



I H -7: 

1750^' 



Art. 51.] COMMON AND PHCENIX COLUMNS. 



AA9 



Common Cohimri. 




P 



36500 



^^^A 



/- 



36500 



2700 d' 



I + 



I' 



P 



36500 



1500 d'^ 



I + 7— :-r. 



1200 d"" 



The formula for " square columns " may be used, without 
much error, for the common chord section composed of two 
channel bars and plates, with the axis of the pin passing 
through the centre of gravity of the cross section. 

Compression members composed of two channels connected 
by zigzag bracing, may be treated by the same formula after 
putting 36,000.00 for 39,000.00 in Eqs. (21) and (22). 



Art. 51. — Experiments on Phoenix Columns,* Latticed Channel Columns 

and Channels. 

In May and July, 1873, some experiments were made at 
Phoenixville, Penn., on full sized Phoenix columns, by the 
Phoenix Iron Co. The results of these experiments are given 
in column headed " Experiment,'' while the column headed 
"/** contains the results of the application of the formula 
established in the preceding Article : 



/ 



42000 



I + 



I 



50000 r^ 



42000 
or = 



I + 



I'' 



(0 



50000 r'- 



* The preceding Article was written as a lecture and read to the Class in Civil 

Engineering at the Rensselaer Polytechnic Institute nearly a year before this Article 

was written ; it has, therefore, been allowed to stand without change. 
29 



450 



PHCENIX EXPERIMENTS. 



[Art. 51. 



according as the ends are " flat " or *' round." All columns are 
"4 segment " ones. 



TABLE I. 



May 3, 1873. 
May 3, 1873. 
May 3, 1873. 
May 3, 1873- 
July 19, 1S73 
July 19, 1873 



ENDS. 


AREA. 


LENGTH. 


r2. 


/2 -5-^2. 


Flat . . . 


Sq. Ins. 

5.84 


Feet. 

23.S1 


4.10 


19.950 


Round.. 


5.95 


24.00 


4.10 


20,230 


Flat . . . 


10.21 


23.3S 


8.68 


9.065 


Flat . . . 


8.50 


22.71 


8.00 


9,282 


Flat . . . 


13-31 


23.20 


8.47 


9.I5I 


Flat . . . 


12.85 


23.20 


8.47 


9.I5I 



Experhnent. 



Lbs. 
30,274.00 

16,387.00 

36,419.00 

38,235-00 

32,742.00 

35,408.00 



Lbs. 
30,000.00 

16,040.00 

35,600.00 

35,430.00 

35,500.00 

35,500.00 



In applying the formula the length was reduced to inches, 
in order to bring it to the same unit as that in which the radius 
of gyration, r, is expressed. 

The columns ^^ Experiment'' and ''/ " are each, of course, 
per square inch. 

It is seen that the experimental results, and those by Gor- 
don's formula, give a very close and satisfactory agreement. 
It is also seen that the analytical relation between flat and 
round ends is a true one. 

The square of the radius of gyration, 4.10, was taken the 
same for the first and second columns because their normal 
sectional areas are so nearly the same. The value 4.10 belongs 
to a 4 segment column, whose area is 5.88 sq. ins. 

The same observation applies to the last two columns. The 
value 8.47 belongs to a column whose area of cross section is 
13.08 square inches. 

A most valuable and instructive set of experiments on Phoe- 
nix columns was also made in the large testing machine at the 



Art. 51.] EXPERIMENTS AT WATER TOWN, MASS. 



4^1 



U. S. arsenal at Watertown, Mass., under the direction of 
Messrs. Clark, Reeves & Co., the results of which were pre- 
sented to the American Society of Civil Engineers at the 13th 
annual convention, June 15, 1881. The value of these experi- 
ments is enhanced by the fact that they were made on full 
sized columns, such in reality as are used in ordinary bridge 
construction. 

In the following table are given the results of these experi- 
ments, as well as those of several formulae presently to be ex- 
plained. 

The following is a portion of the notation : 

/ = length in inches ; 

r = radius of gyration in inches ; 
E. L. = elastic limit in pounds per square inch ; 
Exp. — ultimate resistance in pounds per square inch. 

TABLE II. 



NO. 


LENGTH. 


AREA. 


r2. 


l-^- r. 


/2 + ^2. 


E.L. 


Exp. 


Pv 


/'• 


/"• 




Feet. 


Sq. in. 


Ins. 






Lbs. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


I 


28 


12.062 


8.94 


112 


12,544 




35,150 


32,550 


34,488 




2 


28 


12.181 


8.94 


1X2 


12,54+ 




34, '50 


32.550 


34-488 




3 


25 


12.233 


8.94. 


100 


10,000 


27,960 


35,270 


34,000 


35,040 




4 


25 


12.100 


8.94 


100 


10,000 




35,040 


34,000 


35,040 




5 


22 


12.371 


8,94 


88 


7,744 




35,570 


35,420 


35,592 




6 


22 


12.311 


8.94 


88 


7.744 




34,360 


35,4-0 


35,592 




7 


19 


12.023 


8.94 


7^ 


5,776 




35,365 


36,800 


36,144 




8 


19 


12.087 


8.94 


76 


5,776 


29,290 


36,905 


36,800 


36,144 




9 


16 


12.000 


8.94 


64 


4,096 




36,580 


38,130 


36,606 




10 


16 


I2.000 


8.94 


64 


4,056 





36,580 


38,130 


36,696 




11 


13 


12.185 


8.04 


52 


2,704 


28,890 


36,857 


39,400 


37-248 




12 


13 


12.069 


8.Q4 


52 


2,704 




37,200 


39,400 


37,248 




13 


10 


12-248 


8.94 


40 


1,600 


26.940 


36,480 


40,700 


37,800 




H 


10 


12.339 


8.9- 


40 


1,600 


28,360 


36,397 


40,700 


37,800 




15 


7 


12.265 


8.94 


28 


784 


29,350 


38,'57 


42,200 


38.352 


40,360 


16 


7 


11.962 


8.94 


28 


784 


29,590 


43,^00 


42,200 


38,352 


40,360 


17 


4 


I2.o3l 


8.94 


16 


256 




40,500 


44,770 




46,300 


i3 


4 


12.119 


8.94 


16 


256 


28,050 


51,240 


44-770 




46.300 


19 


8 ins. 


lr.903 


8.94 


2,7 


7.29 




57,130 


69,600 




57,140 


£0 


8 ins. 


11.903 


8.94 


2.7 


7.29 




57,300 


69,600 




57.140 


21 


25' 2.65" 


18.300 


19-37 


68.3 


4,733 




36,010 


37,600 


36,666 




22 


8' 9" 


18.300 


'9-37 


24 


576 


29,510 


42,180 


42,840 




42,160 



In determining r^ for Nos. i to 20, inclusive, a column whose 



452 NEW FORMULA. [Art. 5 1. 

area of cross section was 12.23 square inches was taken. The 
areas of the actual cross sections varied so little from this quan- 
tity, that the resulting value of r^ was assumed to belong to all 
of the first 20 columns. All the columns were tested with flat 
ends. 

An application of Eq. (i) to these columns reveals consid- 
erable discrepancies between the results of that formula and 
the quantities given in the column " Exp'' of the table, when 
the values of / -f- r become comparatively small, as was antici- 
pated in the preceding article. Instead of the constant 42,000 
jn the numerator of Gordon's formula, these experiments show 
that a variable quantity must be used, which shall increase as 
I -^ r decreases, or as r ~ / increases. 

After several trials it was found that the following modified 
form of Gordon's formula would give tolerable results through- 
out the entire range of the experiments : 

40000 ( I + - 

A = \ ~ (2) 




The results of Eq. (2) are given in the column of the table 
headed /i. The agreement between the two columns is not as 
close as could be desired, yet the discrepancies are not sufifi- 
ciently great to vitiate the safe use of the formula. 

In the following figure, the Watertown experiments, as well 
as those of Mr. Bouscaren and the Phoenix Iron Co. (given in 
this and the preceding Article), are shown by diagram. The 
different classes of experiments are indicated as shown. The 
experimental curve is drawn with particular reference to the 
Watertown experiments, for it is then found to be properly 
located in reference to the others. The other curve expresses 
Gordon's formula according to Eq. (2). It would not be diffi- 
cult to find an equation which would fit the experimental 



Art. 51.] 



GRAPHICAL REPRESS. VTA TION. 



453 



curve very closely throughout the range of the experiments, 
but it would not be as simple as Eq. (2), or as two others to be 
shortly given. 

It is interesting and important to observe that each experi- 
mental value in the diagram (which is a mean of two, belong- 
ing to columns of the same length, in the table), lies on or 
exceedingly close to the curve, with the exceptions of those 
shown at a and b. a corresponds to a mean of Nos. 17 and 18, 
and is abnormally high ; b shows the mean of Nos. 13 and 14, 
and is abnormally low. 



140 120 100 80 



60 































































































































































•ri 


w 


ate 


rto 


wr 


Exp 


en 


me 


nt 




















































'f 












































































r 


Bous'cafen 


's 




.' 






















































d 


/, 










1 




























































/i 






13 


Phoe 


nix 








7 
















































( 


1. 


4 








































































■^ //A 


T 




































































y 


^' 




c 














































. 




\a 




.&S- 









rr^ 


p^ 


*=^ 










Cl 




















F 








Ts 


- 






C; 


"o'''<j£!lSJPppT-'': — 


^ 




' 






. 














_ 


^ 


- 


:=: 


= 






— 


Ji>-i 




r=: 


=? 


^= 


^Wat 


ertbwn Exp. Gurye 








b 
























I 
































































































































































































































Fl 


at 


end Ph 


ienix 


Colu 


m 


ns 












































































































































































































































































































bi T 
























1 1 







72000 
60000 
48000 
36000 

24000 
12000 



40 



20 



It may be observed that the experimental curve is nearly a 
straight line from a point just above b to the extreme left of 
the diagram. For that portion of the curve, therefore, the 
following formula applies very closely : 



/' = 39,640 - 46 



(3) 



The results of this formula are given in the column headed 
*'/'." The table, in connection with the diagram, shows that 



454 FORMULA FOR SHORT COLUMNS. [Art. 5 1 

this formula may be used with accuracy for values oi I -^ r 
lying between 30 and 140, and further experiments may pos- 
sibly show that it is applicable above the latter limit. 

For values oi I -^ r less than 30, the following formula will 
be found to give results approximating very closely to the ex- 
perimental curve : 



/" = 64,700 - 4,600 A /- (4) 




The results of the application of this formula are given in 
the column headed ^'/"." 

The extreme simplicity of Eqs. (3) and (4) makes it a mat- 
ter of great interest and importance to determine, by other 
experiments covering extended ranges of / -4- r, whether those 
forms, with different constants, may not apply to shapes other 
than that of the Phoenix column. 

The inapplicability of the true long column formulae, when 
/ 

— is found below certain limits, which is shown in Art. 25, fur- 
r 

nishes a proper foundation for thoroughly empirical formulae, 
such as those expressed in Eqs. (3) and (4). 

By Eq. (4), the ultimate resistance of Phoenix wrought 
iron to pure compression would be about 60,000 pounds per 
square inch. 

The results of the application of Eqs. (3) and (4) to Bou- 
scaren's and the Phoenix experiments are not given, but the 
diagram shows clearly that they would be satisfactory. Data 
sufficient for the application are given in this and the preced- 
ing article. 

The following is the record of the Phoenix tests of the very 
short columns shown at c, d and c in the diagram. It is a ques- 
tion whether the degree of distortion which accompanied the 
extremely high result of 65,867 pounds per square inch, was 
not considerably greater than that which would characterize 



Art. 51.] 



LATTICED COLUMNS. 



455 



NO. 


/. 


AREA. 


r*. 


l^ r. 


n -!- H. 


EXP. 


A- 


/"• 


I 


Ins. 

8 


Sq. in. 
6.9S 


4. II 


3-95 


15.6 


60,573 


51,500 


55,500 


2 


S 


6.98 


4. II 


3-95 


15.6 


60,387 


51,500 


55,500 


3 


4 


5-63 


2.37 


2.6 


6.76 


65,867 


55,800 


57,300 


4 


4 


5-63 


2.37 


2.6 


6.76 


65,867 


55,800 


57,300 


5 


4 


2.93 


2.25 


2.67 


7.13 


56,889 


55,500 


57,200 


6 


4 


2.93 


2.25 


2.67 


7-13 


55,555 


55,500 


57,200 



the condition of '' failure " in an actual structure. This im- 
portant point cannot receive too much attention in connection 
with short column tests, where the relative distortion, in the 
condition of " failure," is far greater than that in long columns. 



Latticed Columns and Channels, 



"^ 



During 1880 and 1881 Col. T. T. S. Laidley, U.S.A., test- 
ed a large number of long columns composed of two chan- 
nel bars latticed in the ordinary manner (Ex. Doc. No. 12, 
47th Cong. 1st Session). These columns were furnished with 
31^-inch pin ends, and were tested at Watertown, Mass., in the 

large government machine. The adjoin- 
ing figure shows the relative positions of 
the channels and pin. 6", 8", 10" and 12" 
Cs were employed, and all the columns, 
the results of whose tests are given in 
Table III., preserved the uniform distance of 8 inches between 
the channels. 

The radius of gyration, r, of the cross section, given in 
that table, is in reference to the axis of the pin. 



45^ 



LATTICED COLUMNS. 



[Art. 51. 



All the posts were single latticed, and the pitch of the 
latticing (the distance apart of rivets in the same flange of a ^) 
was 18 inches for the 6 and 8-inch channels, and 22 inches for 
the 10 and 12-inch. 2" X j^" latticing was used far the 6-inch 
Cs; 2" X ys" for the 8 and lo-inch, and 2%" x ^i" for the 
12-inch. 

The area of cross section for the f^s of the same depth in 
difl"erent columns varied slightly, consequently about an av- 
erage area was taken. 



TABLE III. 
Fin Ends. — 3!^^" Pin. 







AREA OF 


LENGTH IN 


RADIUS OF 


LENGTH OVER 




NO. 


c. 


SECTION IN SQ. 
INCHES (2 CS). 


INCHES. 


GYRATION, INS. 


RADIUS ; OR 

I ^ r. 


/■ 




Inches. 










Pounds. ' 


I 


8 


7.65 


160 


3.00 


53-3 


35,025 


2 


10 


9 


70 


200 


3-65 


54.S 


33,920 


3 


6 


4 


65 


144 


2-35 


C1.3 


34,450 


4 


6 


4 


65 


150 


2-35 


<J3-9 ' 


34,130 


5 


8 


7 


65 


200 


3.00 


66.7 


33,790 


6 


10 


9 


70 


250 


365 


68. 5 


33,770 


7 


6 


4 


65 


180 


2.35 


76.7 


34,iSo 


8 


8 


7 


65 


240 


3.00 


80.0 


32,375 


9 


12 


12 


00 


360 


4.44 


81.0 


31,475 


10 


10 


9 


70 


300 


3-65 


82.2 


33,015 


II 


6 


4 


65 


210 


2.35 


89-5 


31,935 


12 


8 


7 


65 


280 


3.00 


93-3 


31,800 


13 


10 


9 


70 


350 


3-65 


95-9 


30,780 


14 


6 


4 


65 


240 


2.35 


102.2 


30,085 


15 


8 


7 


65 


320 


3.00 


106.7 


29,600 


16 


6 


4 


65 


270 


2.35 


II5-0 


30,820 


17 


8 


7 


65 


360 


3.00 


120.0 


25,885 


iS 


6 


4 


65 


300 


2.35 


127. 8 


24,355 


19 


6 


4 


65 


330 


2.35 


140.6 


21,330 


20 


6 


4 


65 


360 


2.35 


153-4 


15,320 



"/ " is the ultimate resistance per square inch, in pounds. 

All these columns failed as wholes, and each result is 



Art. 51.] 



GRAPHICAL REPRESENTA TION; 



4S7 



a mean of two. Other 
columns of the same set, 
and tested at the same 
time, failed by buckling 
of the channels ; they 
cannot, consequently, be 
classed among long col- 
umns which are so con- 
structed as to fail as 
ivholcs. 

The values of / in 
Table III. are shown 
graphically in Plate I. 
The ratio I ~ r is laid 
off along the horizontal 
line and the ultimate in- 
tensity/ on the vertical 
line, as shown. The full 
curved line is then the 
experimental curve and 
possesses great value of 
a practical nature. 
Within the limits of the 
diagram, when the ratio 

/-- r 

is known, the ultimate 
resistance of the column 
per square inch (/) can 
be at once accurately 
read from the plate 
without calculation or 
scale. 

The following equa- 
tion : 









r 















p 








— 







-r 


— 


n 




~ 


— 







"■ 


— 


-r 






~^ 


• 




































































^ ' 





























































, 








fl) 









V 































































•♦-' 


•* 






I 






ro 
















CN 


















V" 




















CO 




































































1 




IL 










■ 






























































-If* 










































































~Tr 

























































































1 
































































vu 




































1 




1 


1 








































































1 










































1 




























1 












1 




























, 






























1 








































































1 








































































1 






































c 




































1 


































1 










































































t^ 







































































































1 
































J> 
































1 




















1 


















































































































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1 






























1 






































































































1 








































































ll\ J 






































1 










1 


















1«-ll 






































1 
































ITvi- 




































1 
































n V 




































1 


















.0 






































































CO 














1 Wj 








































































11 ^ 


























































- 














JM V- 






































































r 1 
































1 


















































































1 1 




































































































1 


























































































































































!l 


























































D 
















u\ 
























































CTS| 
















\ 


1 












































\ \ 


























\ 


1 












1 
































1 




























% 
































































1 












































































































1 




















































IL, 1 






































































i I 










































































1 \ 
























































ol 
















1 \ 






































„ 
















ol 












1 \ 




\ 


1 






























1 


1 


















— 
















\ 


\ 


1 


































1 






























1 


I 


































1 


















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1^ 


































T3 





































































































1 




















( 



































! 
















V 
















1 












•r; 


















I 


















: 


V 






























tL 


















i 


















' 


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n1 


















_o 


















\ 












1 








































1 














i ' 










1 






















1 








r- 




1 




1 










\ 1 










1 




1 


















































\ 
















1 












i* 






































\ 












~1 


















■H 


































1 


i\ 


1 




























... 






























1 




. 


|\ rr 
























n 




























1 




^ 


V 1 






























1 




1 


































\ il 








































































1 
































































































\- 


1 


1 








CN 






























1 




















1 
































v 

































































































































\ 







































































v 




























































































1 




I 














































t 










1 1 








i ; 












































\ 














1 










111 










































\ 


X 












1 








































































c 














ro 


















































.E 














» 


— 


















































n 




































































\ 


" 
















- 
















































^ 


0, 










































lA 








1 










H\ 












































\i 


1 








1 






1 


CO 


































































1 
















- 


















































1 




















- 


























\ 






















1 








































'. ^ 


















1 


1 














+ 




























\ 


1 














1 
















v-| 






















1 








^-M 


































" 






























iTT 






























































J 1 1 


















1 


















































^ 


1 








■ 


1 1 






























1 






























i 


i 1 














































1 






\l 










1 










1 






































1 






\ 




























1 


































\ 






"1x1 


























(1 


-1 


































\ 






\ 




























n 
















1 


1 












~^ 






\ 
























• 


— 




; 
































\ 
















_ 














































"^ 
























1 


































\ 






1 






















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■V 






1 


















n 








-J 






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1 1 






1 




■■ ■ 1 l-\ 




1 ! 












_. 





458 



LATTICED COLUMNS. 



[Art, 51. 



/ = 



39000 



I + 



__i i^ 

30000 r"" 



(5) 



probably gives as accurate results as any form of Gordon's 
formula. The dotted curve is constructed from it. Its re- 
sults are seen to be only tolerably approximate between the 

limits — =^ 50 and 135. It possesses little value when com- 
pared with the plate. 

Table IV. contains results for columns of the same set 
which failed by buckling of the individual channels of which 
they were composed. 

TABLE IV. 



NO. 


c. 


LENGTH, 
INCHES, 


RADIUS OF 

GYRATION IN 

INCHES, r. 


/ 
r 


ULT. / IN LBS. PER 
SQ. INCH. 


CONDITION 


OF ENDS. 


I 


Inches. 
6 


120 


2.35 


5I-I 


36,025 


Flat. 




2 


6 


120 


2.35 . 


5I-I 


33,740 


One flat ; 


one pin. 


3 


10 


126 


3-65 


34-5 


35,450 


Pin. 




4 


12 


120 


4.44 


27.0 


34,245 


Pin. 




5 


12 


180 


4.44 


40.5 


34,660 


Pin. 




6 


12 


240 


4-44 


54-0 


33,985 


Pin. 




7 


12 


300 


4-44 


67.5 


33,590 


Pin. 





If r is the radius of gyration in reference to an axis 
through the centre of gravity of a single channel section, and 
parallel to the weby the following values will hold for the pres- 
ent cases : 



Art. 51.] BUCKLING OF LATTICED COLUMNS. 



459 



6"C 

8" C 

10" C 

12" c 



?*' z=L 0.58 inches, 
r' = 0.48 inches. 
r' = 0.69 inches. 
r = 0.87 inches. 



Although the lattice rivets were alternate in the same chan- 
nel, each flange was unsupported for a distance equal to the 
pitch, ie., 18" for the 6" and 8" Cs, and 22" for the 10" and 
12" Cs. Hence : 



For 6" C ; 


18 - / 


= 31-0 


For 8" C ; 


18 -f- r' 


= 374 


For 10" C ; 


22 - / 


= 31-9 


For 12" C ; 


22 -^ r 


= 25.3 



Table IV. shows that the column of 10" Cs commenced to 
fail by buckling of the j^s when 



and when 



I -^ r = 34.5, 



22 4- r = 31.9; 



that the column of 12" Cs commenced to fail similarly when 
the lenp-th' became so small that 



while 



I ^ r — 27.0, 



22 ~ r = 25.3. 



These results would seem to show that pin end columns 
with single but alternate latticing will begin to fail by buck- 
ling of the channels when I -i- r, for the column as a whole, 
becomes so small that it is about equal to the same ratio for a 
single channel between two adjacent rivets in the same flange. 



4^0 LATTICED COLUMNS, [Art. 5 1. 

Nos. I and 2 of Table IV. show that if the ends possess a 
greater degree of fixedness, the value of / -f- r is much greater 
when buckling begins to take place, but the number of experi- 
ments is not sufficient to indicate the exact amount. 

As would be anticipated under the circumstances,/ main- 
tains about the same value for all the columns in Table IV. 
Hence when I -^ r becomes so small that buckling takes place, 
the ultimate resistance of the column is independent of the 
length. 

The graphical representations of the results given in this 
Article show that the curve of ultimate resistances has a very 
sharp declivity for small values of /-f- r, but that it becomes 
nearly straight and horizontal for larger values, and that it 
again increases in declivity with a still father increase in that 
ratio. These phenomena seem to be much more pronounced 
in the tubular variety of columns. They find a simple and 
obvious explanation in the fact that in columns of moderate 
length the deflection at the centre of the column about keeps 
pace (in the same direction) with the movement of the centre 
of pressure at the ends. 

Plate I. shows (what was to be anticipated) that this effect 
is also much less pronounced with pin ends than with flat 
ones, it being borne in mind that the phenomena here consid- 
ered do not produce the horizontal straight line which would 
be seen if Plate I. included less values of / -f- r than 50. The 
latter represents the buckling of the individual parts of the 
column, and not the failure of the column as a whole. 

A few experiments by Col. Laidley with columns of the 
same Cs as the above, but with pins only three inches in diam- 
eter, gave uniformly less ultimate resistance than those with 
three and a half inch pins. Although this result was to be 
expected, the number of experiments was not sufficient to 
justify any quantitative conclusions ; it can only be stated 
that the smaller the pin the less will be the ultimate resist- 
ance. 



Art. 51.] 



FLAT END CHANNELS. 



461 



TABLE V. 
Flat End ^s. 







AREA OF 








i;lt. resist., in 


NO. 


c 


SECTION IN SO. 
INCHES. 


/. 


r' . 


r' ' 


IBS. PER SQ. INCH 




Inches. 




Inches. 


Inches. 






I 


6 


2.33 


6.00 


0.58 


10.3 


42,293 


2 


6 


2.33 


17.60 


0.58 


30-3 


36,835 


3 


6 


2.33 


23 90 


0.58 


41. 1 


33,910 


4 


6 


2.33 


48. 00 


0.58 


82.6 


28,140 


5 


8 


3. So 


8.00 


0.48 


16.6 


43,295 


6 


8 


3 -80 


17.90 


0.48 


37-2 


35,280 


1 


8 


3.80 


23.90 


0.48 


49 7 


35,975 


8 


8 


3-8o 


29.90 


0.43 


62.2 


33,4co 


9 


8 


3.80 


48.00 


0.48 


99.8 


30,620 


10 


10 


4-85 


10.00 


0.69 


14-5 


35,080 


II 


10 


4-85 


17.90 


0.69 


26.0 


33,820 


12 


10 


4.85 


23.90 


0.69 


34-7 


34,355 


13 


10 


4.85 


29.90 


0.69 


43-4 


34,050 


14 


10 


4-85 


48. CO 


0.69 


69.6 


54,oSo 


15 


12 


6.00 


12.00 


0.87 


13.8 


37,240 


16 


12 


6.00 


17.80 


O.S7 


20.5 


36,590 


17 


12 


6.00 


23 . 90 


0.87 


27-5 


36,695 


iS 


12 


6.00 


29 90 


0.87 


34-4 


35,150 


19 


12 


6.00 


48. 00 


0.87 


55-2 


36,040 



Table V. contains the results of Col. Laidley's tests of por- 
tions of the Cs used in the columns which have just been 
treated. These portions had flat ends. 

The moment of inertia of the section, from which the 
radius of gyration r was computed, was taken about an axis 
parallel to the web of the channel and passing through its 
centre of gravity. 

Many of the results are means of two tests each. 

The results given in Table V. are shown graphically in 
Plate II. The values of the ratio I -^ r are laid off on the hori- 
zontal base line, to the left from O ; while the values of / in 



462 



CHANNELS AS COLUMNS. 



[Art. 51. 





TTTT 








"TT 


"" 




■"" 


1 7^ 


i 1 1 






T 


TFT 


■" 


""J— 


^ 


c 




« 


I'lo 
























1 1 






1 


<^ 


















•-• 


1 1 
























1 i 




1 1 























1 ! 


1 











i 1 









1 1 


1 


] 1 1 


















f? 


•<r 


^ Ll 




fO 


1 1 




CN 




1 1 


1 


1 1 


^ 


1 








1 






Tn 


1 




1 




















1 












1 








1 






1 






1 














1 






1 




1 






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1 














1 














1 






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1 


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1 




















1 






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1 


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IN 


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r- 


















- 
























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pounds per square 
inch are laid off ver- 
tically from O, as 
show n. The full 
curve then repre- 
sents with great ac- 
curacy the experi- 
mental results. 

The dotted curve 
represents the fol- 
lowing form of Gor- 
don's formula for 
the ultimate resist- 
ance in pounds per 
square inch : 



/ = 



_ 36000 



1 + 



l' 



(6) 



63000 r' 



This formula is 
sufificiently accurate 
for all ordinary pur- 
poses, between the 
limits 



/■r- r' 



15 



and 



/ -^ r' = 90, 

but does not com- 
pare in value with 
the experimental 
(full) curve. 



Art. 52.] LONG COLUMN FORMULA. 463 



Art. 52. — Euler's and Tredgold's Forms of Long Column Formulae. 

T\i^ form of the general formula given in the preceding Ar- 
ticle, as will presently be shown, does not seem to be as well 
adapted to the expression of. accurate results as that of Euler, 
given in Art. 25. 

It has already been observed that the coefificient a (Eq. (9) 

/>'■' 
of Art. 50), contains -^ as a factor, in which p" is the great- 

est intensity of bending stress, /.r., a part of the quantity ''/ *' 
which is sought. The possible use of the formula is based on 
the fact that E is very large in respect to/". 

The existence of/" in a is due to the redundant form of Eq. 
(8) of the Article cited. 

Article,/' = 

(7)), Eq. (8) gives : 



P act) 

Since, in that Article,/' = — and <^ — a^c' = ■ (see Eq. 



^ ~' "" Sr^~ E Sr^ ' 



P E r' J r' , , 



This is Euler's formula as given in Eq. (6) of Art. 25. In 
this equation b has the analytical values A,n^E, n^'E and 2.25 tv'E 
for ends fixed, rounded and one fixed one rounded, respect- 
ively, as shown in Art. 25. 

It would seem, therefore, that, since Eq. (i) involves noth- 
ing variable in the second member but r ~- l,\\. ought to give 
more accurate results than Tredgold's form of Art. 50. 

It was shown, however, in Art. 25 that the common the- 
ory of flexure is analytically applicable only to fixed end col- 
umns of wrought iron, in which the ratio of length over radius 



4^4 EULEKS FORMULA. [Art. 52. 

of gyration is somewhat greater than 140; and to round 
end columns in which that ratio is somewhat greater than 
70. Since the implicit assumption of an indefinitely small cross 
section underlies the analytical treatment of long columns, it is 
possible that the analytical coefficients and exponent may not 
obtain far above the limits indicated in Art. 25. Now, since 
other conditions of ends will lie between these limits, it is seen 
that both long column formulae are strictly inapplicable to a 
large portion of the columns designed by engineers. 

Fortunately, a sufficient number of experiments have been 
made with full sized columns to show that either y^?^;;^ of for- 
mula, when holding empirical quantities properly determined, 
will give excellent results. This has already been shown for 
Tredgold's form, and it will now be seen that Euler's form may 
be expected to give still better results. 

If, as is usual, r is the radius of gyration and / the length 
(both in the same unit), and if both the coefficient and exponent 

of -y , in Euler's general formula, be considered variable, the 

following equation (see Art. 25), may be written : 

For other values (/ and /') of r and /, the mean intensity 
becomes : 

/=^(7'y (3) 

Dividing Eq. (3) by Eq. (2), then taking logarithms and 
solving for x : 



^=r7^ ('^^ 



Art. 52.] RESULTS FOR PHOENIX COLUMNS. 



465 



Subtracting Eq. (3) from Eq. (2) and solving for 7: 






/' 



(5) 



These formulae will first be applied to results of the experi- 
ments made on Phoenix columns at Watertown, Mass. These 
results are contained in Table II. of the preceding Article, and 

the columns and " Exp.'' are reproduced in Table I. of this 

r 

Article. In the latter, however, the column ** Exp'' contains 

the means of the various pairs of experiments whose results 

are given in the former. 

TABLE I. 
Phcenix Columns. 



I 

r 


Exp. 


P- 


r 


Exp. 


P- 


112 


34.650 


34.550 


40.0 


36,440 


39.000 


100 


35.150 


35>ooo 


28.0 


40,700 


40,630 


88 


35,000 


35.530 


16.0 


50,400 


43.400 


76 


36,130 


36,150 


2.7 


57,200 


53.400 


64 


36,580 


36,900 


68.8 


36,000 


• 
36,570 


52 


37.000 


37,800 


24.0 


42,200 


41,400 



Now, let there be taken : 



1= 28 
r 



30 



p = 40,700. 



466 RULER'S FORMULA. [Art. 52. 

I' 

-7 =112 p = 34,650. 

r 

Inserting these values in Eqs. (4) and (5), there will result : 
X = 0.1 17 and J/ = 59,723. 

Then let there be written : 

/^\ 0.T17 
/ = 60,000 [jj (6) 

The various values of f — j in Table I., inserted in Eq. (6), 

give the results shown in columns **/ " of that Table. They 
are seen to be much more satisfactory, as a whole, than those 
given by any form of Tredgold's formula in the preceding 
Articles; although Eq. (2) of Art. 51 is a little closer to the 

experimental results for values of — less than 24. 

r 

So much of the curve represented by Eq. (6) as does not 
coincide with the experimental curve, is shown by the dotted 
line in the Fig. of the preceding Article. 

That curve, together wittw the results given in Table I., 

shows the close agreement of Eq. (6) with experiment for all 

^ I 

values of -y from i to — . 
/ 112 

It is interesting and important to observe that when — = i, 

Eq. (6) gives : 

/ = 60,000 ; 

or about the ultimate compressive resistance of wrought iron 
in cubes. 

An application of Eqs. (4) and (5), in the manner already 



Art. 52.] RESULTS FOR KEYSTONE COLUMNS. 



467 



shown, to the results of Bouscaren's experiments on Keystone 
columns, given in the large table of Art. 50, gave the following 
results for swelled Keystone columns : 



X = 0.25 and J/ = 78,000; or 



/ = 78,000 ( J 



(7) 



TABLE II. 
Keystone Columns. 



SWELLED. 




STRAIGHT. 


• 


C. 


Exp. 


/. • 


c. 


Exp. 


/. 


326 


33,600 


37,800 


8,718 


25,000 


28,000 


2,991 


28,800 


28,700 


9,391 


27,500 


27,700 


9,646 


24,100 


24,800 


9.157 


30,000 


27,800 


3,130 


36,900 


28,500 


3,519 


30,000 


31,350 


9,189 


21,100 


24,900 


4,136 


32,000 


30, 700 


9,157 


25,400 


24,900 


10,714 


27,800 


27,300 



Also, for straight Keystone columns : 

X = 0.25 and f = 87,000 ; or 



/ = 87,000 



(8) 



The results of the application of these formulae, and the ex- 



4^8 EULER'S FORMULA. [Art. 52. 

perimental results, are given in Table II. The lengths and 
other data can be found in the table just cited. 

By the same operations with the square column results 
(Bouscaren's) of the same table, there were found : 

X = 0.5, and J = 303,000; or: 



/ = 303,000 f-V ..... .. (9) 



The following columns, ^' ExpT and *'/ " contain the ex- 
perimental square column results and those computed from 
Eq. (9). 

• c. Exp. /. 

'°'4I4 30,000 30,000 J 3^^^^ 

7-^33 33,200 33,000 U^j^_^_^^^ 

9,623 30,200 30,600 ) 

Only " flat end " experiments have been treated, for the 
others are utterly insufficient in number for the determination 
of the empirical quantities. 

In fact, with the exception of the Watertown experiments 
on the Phoenix columns, the number of those with *' flat ends " 
is not sufficiently great, nor the range oi I -^ r sufficiently ex- 
tended, to establish reliable formulae. 

In all cases, however, it is to be observed that the formulas 
of this Article give results more nearly agreeing with the ex- 
perimental ones than those computed from any form of Tred- 
gold's or Gordon's formula. It would seem that this form of 
formula has not heretofore received the attention to which its 
importance and value entitle it. 

Each of the three Eqs. (7), (8) and (9), become inapplicable 

when the value of — is such that ^^ p " approaches the ultimate 



Art. 53.] HODGKINSON'S FORMULA. 469 

compressive resistance per square inch of wrought iron in 
short blocks. 

These empirical results tend to give experimental confirma- 
tion to Euler's formula, for the exponent and coefficient of ( — ] 

are seen to increase very much as the lowest value of ^, in the 
different sets of experiments, increases. 



Art. 53. — Hodgkinson's Formulae. 

The detailed account of the experiments on which Eaton 
Hodgkinson based his various formulae is given in the Phil. 
Trans, of the Royal Society of London, for 1840. His cast-iron 
columns were small ones, the greatest length of which was 60.5 
inches. The greatest value of the length divided by the radius 
of gyration was : 

- = 2 X — ^ = 484 ; 
r 0.25 ^ ^ 

while the least value of the same ratio was : 

7 "" ^ ^ 0:5 "" ^^'^ (nearly). 

The greatest diameter was about two inches. 

Let d = diameter of column in inches. 
Let / = length of column in feet. 

Then for the breaking weight (P) of solid cylindrical cast- 
iron columns, when expressed in pounds, Hodgkinson's for- 
mulae take the shape :. 



4/0 HODGKINSON'S FORMULA. [Art. 53. 

^376 

P = 33»379 777 ; (foi* rounded ends) . . . (i) 

^3-55 

P = 98,922 -jYY ; (for fixed ends) .... (2) 

For hollow cylindrical columns of cast iron : 

P= 29,120 -r;^ ■ ; (for rounded ends) . . (3) 

P = 99,320 jY^ ; (for fixed ends) ... (4) 

In Eqs. (3) and (4), D is the greater, or exterior, diameter 
of the column, while d is the interior diameter. It is to be 
observed that P is the total breaking weight in pounds. 

The longest wrought-iron solid cylindrical column tested 
by Hodgkinson had a length of 90.75 inches and a diameter of 
about 1.02 inches. Hence the greatest ratio of length over 
radius of gyration was about 90.75 X 4 = 363. 

His formulae for the total breaking weight of solid cylindri- 
cal wrought-iron columns, in pounds, are : 

<^3 76 

P = 95,848 —7^ ; (for rounded ends) ... (5) 



^3-55 

P = 299,617 -Tj- ; (for fixed ends) .... (6) 

In his experiments on square pillars of Dantzic oak, the 
greatest dimensions were : length = 60.5 inches, and side of 
square section = 1.75 inches. 

His longest red deal pillar was 58 inches in length, and the 
cross sections were 1x1,1x2 and i x 3 ; all in inches. 



Art. 53-] TIMBER COLUMNS. 47 1 

Hodgkinson used Lamande's experiments on French oak 
in establishing a formula for that material. In those experi- 
ments, the longest pillar had a length of 76.5 inches and a 
normal section of 2.13 inches by 2.13 inches. 

Retaining the same notation, the following are the total 
breaking weights, in pounds, of solid square timber pillars with 
flat ends : 

Dantzic oak (dry) ; P = 24,542 7^ • • • (7) 



Red deal (dry) ; P=. 17,511^ ... (8) 



^4 
French oak (dry) ; P = 15,455 _- . . . (9) 

In Eqs. (7), (8) and (9), " d'' is the side of the square sec- 
tion of the column in inches, while /is the length in feet. 

All the preceding formulae are to be used only in those 
cases in which the length exceeds 30 times the diameter or 
side of square, if the ends are fixed ; or 15 times the length, if 
the ends are rounded. Between these limits and a short block, 
in which the length is 4 or 5 times the diameter or less, the 
following formula is to be used : Let C be the ultimate com- 
pressive resistance of the material, per unit of area, in short 
blocks, and let A be the area of the normal section of the col- 
umn ; then Hodgkinson's formula for these columns of inter- 
mediate lengths is : 

r>' — PC A 

^ ~ P+^4CA <'°> 

The small size of the columns experimented upon by 
Hodgkinson militates very strongly against the practical value 
of his formulae, unless it should be shown experimentally that 



4/2 HODGKINSON'S CONCLUSIONS. [Art. 53. 

the same formulae may be equally applicable to large and small 
columns. 

With the greatest ratio of / over r, the ratio of the resist- 
ance of a fixed end pillar over that of one of the same length 
and with rounded ends was about 3.34. With the lowest value 
of / over r, the same ratio was about 1.63. According to Eu- 
ler's formula, that ratio should have been 4. It is seen, there- 
fore, that with these columns the common theory of flexure 
failed far above the limit given in Art. 25. 

From his experiments Hodgkinson drew the following con- 
clusions : 

The strength of a pillar with one end round and the other 
flat, is the arithmetical mean between that of a pillar of the 
same dimensions with both ends rounded, and with both ends 
flat. 

A long uniform pillar, with its ends firmly fixed, whether 
by disks or otherwise, has the same power to resist breaking as 
a pillar of the same diameter and half the length, with the 
ends rounded or turned so that the force would pass through 
the axis. 

Long uniform cast-iron pillars with both ends round break 
in one place only — the middle ; those with both ends flat, near 
each end and at the middle ; those with one end round and 
one end flat, about one-third the length from the round end. 

The resistance of solid pillars with round ends was increased 
about one-seventh by increasing the diameter at the middle. 
Flat-end pillars (solid) had their resistances increased very 
slightly by the same means, but hollow pillars seemed to derive 
no benefit at all by enlargement at the middle. 

The resistance of flat-end pillars was increased slightly by 
the application of disks to their ends. 

Irregular and imperfect fixedness of the ends may cause a 
loss of two-thirds, or more, of the resistance with ends per- 
fectly fixed. 

Solid square cast-iron pillars failed in diagonal planes. 



Art. 54.] 



TUBES AS COLUMNS. 



A7^ 



The relative resistances of columns of the same length and 
area of cross section were about as follbws : 

Long, solid, round pillar 100 

** " square pillar 93 

*' ** triangular pillar no 



Art. 54. — Graphical Representation of Results of Long Column 

Experiments. 

If the values of / over r (length over radius of gyration), for 

TABLE L ' 

Tubes. — Flat Ends. 















ULT. RESIST. PER 


NO. 


LENGTH. 


EXT. DIA. 


THICKNESS. 


AREA. 


/-5- r. 


SQUARE INCH. 




Inches. 


Inches. 


Inch. 


Sq. ins. 




Pounds. 


I 


120 


1-5 


O.IO 


0.44 


240 


14,670 


2 


120 


2.00 


0. 10 


0.61 


179 


23,206 


3 


120 


2.35 


0.23 


1.50 


160 


2Ij900 


4 


120 


2.50 


O.II 


o.So 


141 


29,800 


5 


120 


3.00 


0.15 


1-35 


120 


27,670 


6 


60 


1.50 


0. 10 


0.44 


120 


31,180 


7 


go 


3-04 


0.17 


1. 41 


90 


29,790 


8 


60 


2.00 


0. 10 


0.61 


89 


33,300 


9 


120 


405 


0.16 


1.9 


87 


26,960 


10 


60 


2.35 


0.22 


1.47 


80 


29,330 


11 


60 


2.34 


0.21 


1-37 • 


80 


30,000 


12 


60 


2.50 


0. II 


0.80 


71 


35,100 


13 


89 


4.00 


0.24 


2. 87 


67 


26,800 


14 


90 


4-05 


0. 12 


1. 61 


65 


33,330 


15 


30 


1.50 


O.IO 


0.44 


60 


34,220 


16 


60 


4.00 


0.24 


2.85 


45 


32,200 


17 


30 


2.00 


O.IO 


0.61 


45 


36,980 


18 


30 


2.35 


0.24 


1.60 


40 


35,660 


19 


30 


2-34 


0.21 


1.44 


40 


36,000 


20 


29 


2.37 


0.23 


1-55 


39 


36,910 


21 


29 


2-34 


0.20 


1.36 


39 


39,570 


22 


30 


2.50 


O.II 


0.80 


35 


36,490 


23 


28 


3-00 


0.15 


1. 41 


28 


37,390 


24 


28 


4.00 


0.25 


2.85 


21 


48,200 



474 



GRAPHICAL REPRESENTA TION. 



[Art. 54. 



a series of columns which have been tested to breaking, be 
accurately laid off on a horizontal scale, and if the breaking 
weights per square inch be laid off with equal accuracy on a 
vertical scale, the resulting curve will represent the resistances 
of all columns for which / over r lies within the limits of the 
experiments, v/ith far more accuracy than any simple and prac- 
ticable formula that can be devised. Such a curve for the 
Watertown experiments on Phoenix columns has already been 
incidentally constructed in Art. 51. 

TABLE II. 

Solid RccianfTiilar Pillars. — Flat Ends. 













ULT. RESIST. PER 


NO. 


LENGTH. 


SECTION. 


AREA. 


l-^r. 


SQ. IN. 




Inches. 


Inches. 


Sq. Ins. 




Pounds. 


I 


120 


2. 98 X 0.5 


1-5 


822 


8,160 


2 


90 


2.98 X 0.5 


1-5 


643 


2,410 


3 


120 


3.01 X 0.77 


2.31 


540 


3,380 


4 


120 


3 . 00 X 1 . 00 


3.00 


414 


4,280 


5 


60 


2.98 X 0.5 


1.50 


400 


5,630 


6 


90 


(5.86 X 0.99 ) 
] 3.00 X 1. ) 


3.00 


311 


9,600 


7 


90 


1 . 02 X 1 . 03 


1.05 


300 


9,750 


8 


120 


3 . 00 X 1 . 5 1 


4-53 


272 


10,170 


9 


60 


3.01 X 0.77 


2.31 


270 


12,970 


10 


60 


3.01 X 0.99 


2.99 


207 


18,070 


II 


60 


5.84 X 1. 00 


5.84 


207 


17,700 


12 


30 


2.99 X 0.50 


1.50 


206 


16.850 


13 


90 


3.00 X 1.53 


4-59 


204 


19,990 


14 


60 


1.03 X 1.02 


1.05 


200 


17,270 


15 


30 


3.01 X 0.76 


2.30 


135 


27,770 


16 


30 


3 . 00 X 1 . 00 


3.00 


104 


29,660 


17 


30 


1 . 02 X 1 . 02 


1 .04 


100 


25,330 


18 


15 


1.02 X 1.02 


1 .04 


50 


34,550 


19 


7-5 


1.02 X 1.02 


1.04 


25 


48,680 


20 


3-8 


1.02 X I 02 


1.04 


13 


50,400* 



Bore this without failure. 



Tables I., II. and III. contain the results of some English 
experiments on small flat-end wrought-iron columns of different 



Art. 54.] 



FLAT END COLUMNS. 



475 



- 




1 1 


"1 M 


~ 


— 


1 i 


^rr 


~T- 


-'■TIT 


I' 1 


~r" 


Mil 


"" 


- 


1 1 




> 


1 


-■\T 


MM 


1 1 i 


~ 


— 1 


-rr 


~7~ 




1 1 


1 








1 1 




1 


1 


i 1 1 




1 


M !.i. 








1 


1 


1 


1 


1 




Mil 


1 1 






1 1 


! 






1 i 


1 


1 






1 1 




1 


1 


! t 




! 


1 


1 1 












1 


1 


1 






1 


1 1 








1 


1 








," 


I 




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1 


1 


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1 1 








! 


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c 




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r~i 


1 




















< 




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or 


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B 


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1 




















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c 






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in 




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1 


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= 


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1 



















All these experiments were on small cross sections. 
Were little more than models. 



In reality the columns 



476 



GRAPHICAL REPRESENTA TION. 



[Art. 54. 



forms of cross section. These results are taken from the '* Pro- 
ceedings of the Institution of Civil Engineers," of London, Vol. 
XXX. The experiments on tubular and angle-iron columns 
(Tables I. and III.) were made by Mr. Davies, while those on 
solid rectangular columns (Table II.) were made by Mr. Hodg- 
kinson. The graphical representation of these results is shown 
by a very accurate construction in Plate III. Fig. i belongs 
to Table I. ; Fig. 2 to Table II. ; and Fig. 3 to Table III. The 
result shown at a (No. i of Table II.), Fig. 2, is most anoma- 
lously high, as is very evident, and has been neglected. 

The horizontal scale shows the ratio of / over r, while the 
vertical scale shows pounds per square inch, to a scale of 
30,000.00 pounds to the inch. 

TABLE III. 

3" X 3" X 1% \^s.—Flat Ends. 











ULT. RESIST. PER 


NO. 


LENGTH. 


AREA. 


l^ r. 


SQ. IN. 




Inches. 


Sq. Ins. 




Pounds. 


I 


60 


1. 73 


71 


23,600 


2 


48 


1. 73 


56 


29,480 


3 


36 


1.73 


42 


35,380 


4 


iS 


1.73 


21 


39,400 



There are only four results with angle irons, but so far as 
they extend, they are less, for the same value of / over r, than 
those for either tubes or solid rectangular sections. This was 
to be expected, since the legs of angles are comparatively thin 
and give very little lateral support to each other. A single 
unsupported angle iron, therefore, does not make a good com- 
pression member. 



Art. 55.] ANGLE IRON COLUMNS. 47/ 

These results, in connection with those of Art. 51, show 
very clearly that an empirical curve (or formula) may be con- 
structed to cover, with sufficient accuracy for practical pur- 
poses, columns of different forms of cross section, provided they 
are so built that their component parts are mutually supporting. 

As compression members of single angle irons with fixed 
ends are quite common in some riveted bridge and roof trusses, 
it would be desirable to frame a formula on an extended series 
of numerous experiments. In the present instance this is im- 
possible, but the following formula may be used with safety 
for equal legged angle iron columns with flat or fixed ends, so 
long as / -^ r lies between 20 and 100: 



/ = 200,000 A /- (l) 




in which/ is the ultimate resistance per square inch. An ap- 
plication to the columns of Table III. gives the following re- 
sults : 

No. / -f- r /. 

1 71 23,740 lbs. per square inch. 

2 56 26, 760 lbs. per square inch. 

3 42 30,860 lbs. per square inch. 

4 21 43,600 lbs. per square inch. 

By comparison with the results in Table III., the deviations 
from the actual resistances given by experiment may be seen 
at a glance. 

Art. 55. — Limit of Applicability of Euler's Formula. 

The great range of / -^ r in the experimental results of 
Tables I. and II. of the preceding Article, furnishes means of 
testing the applicability of Euler's formula with high values of 
that ratio. 



478 



LIMIT FOR RULER'S FORMULA. 



[Art. 55. 



Mr. Hodgkinson determined the mean value of the com- 
pressive coefficient of elasticity for some wrought iron of pre- 
sumably the same grade as that to which Table II. belongs, at 
about 23,250,000 pounds per square inch. That value gives : 



^n^'E = 917,920,000. 



Taking I -- r from No. i. Table I. : 



/ = 47r=^(-j = 16,000 (nearly) 
Experiment gave 14,670 



. . (2) 



Taking I -^ r from No. 7, Table II. : 



/ — 47r''Ei-j) = 10,200 (nearly) 
Experiment gave 9,750 



(3) 



Taking I -^ r from No. 5, Table II. : 



/ = 4^'^(^-^y = 5,740 (nearly) 
Experiment gave 5,630 



(4) 



Taking I -^ r from No. 3, Table II. : 



/ = A-^^Ei-j) = 3,150 (nearly) 
Experiment gave 3,380 



(5) 



Art. 56.] REDUCTION OF END SECTION. Al9 

Taking I ~ r from No. 2, Table II. : 

sarly) 

^ .... (6) 



p z=z 47r^E(—\ =z 2,220 (nearly) 



Experiment gave 2,410 

In Eq. (2), / -4- r is 240, yet the result by formula is 
only a little too large. With / -^ r ranging from 300 to 643, 
the formula gives very satisfactory results. These tests would 
seem to show, therefore, that only when / -^ r becomes equal 
to about 250 for flat-end columns, does Euler's formula become 
applicable to wrought-iron compression members, but that 
above that limit it gives very satisfactory results. 

This is an interesting and striking confirmation of the cor- 
rectness of the formula, which, as was stated in Art. 25, is 
based on the supposition that the lateral dimensions are very 
small compared with the length. 

Art. 56. — Reduction of Columns at Ends. 

When columns are built of angle irons, channel bars, or I 
beams, it is frequently the practice to cut off, for some distance 
back from the ends, the flanges of bars or beams, or one of the 
legs of angle irons, in order to give clearance for other mem- 
bers of the structure. In such cases the whole compression to 
which the column is subjected is carried, at the ends, by the 
webs of the bars or beams, or legs of the angles, which are thus 
solid rectangular columns of great comparative breadth and 
little thickness, even when reinforced by plates of the same 
thickness as the webs or legs. In such cases, the angle iron 
experiments of Mr. Davies (a part of which are given in Art. 
54), and a most valuable set of full sized, latticed, channel- 
bar column tests, made at the works of the Keystone Bridge 
Co., Pittsburgh, Penn. (" The American Engineer," 4th Feb., 



4Bo TIMBER COLUMNS. [Art. 57. 

1882), show that the full resistance of the column is not devel- 
oped, but that they fail at the ends where the cutting away of 
the flanges and legs reduces the column to two thin, weak, rect- 
angular columns. Columns, therefore, should never be cut 
away in the manner indicated unless the circumstances render 
it absolutely necessary, and then the ends should be reinforced 
by extraordinarily heavy thickening plates, so that the sum of 
the resistances of these rectangular columns, at each end, shall 
be equal to that of the column as a whole. 

Art. 57. — Timber Columns. 

Tests of this class of members, the results of which have 
been published, although of great value, have not been made 
with sufficiently large ratios of length to radius of gyration to 
produce true " long column " failures. This renders impos- 
sible the establishment of a long column formula or diagram 
for practical use in connection with the use of long timber 
columns. 

Some very valuable experiments, however, have been made 
with full sized columns having lengths as great as fourteen 
feet. The first results to be given are those of a large number 
of tests by Prof. Lanza, of Boston, in which he used the United 
States testing machine at Watertown, Mass. These tests were 
made during 1881, on such members as are commonly used in 
the construction of cotton and woollen mills. 

Table I. contains the results of Prof. Lanza's tests. A large 
majority of the columns had cores bored out of the centre, 
which varied in diameter from 1.5 to 2.0 inches. The ab- 
sence of material did not affect, in any way, so far as could be 
observed, tHe resistance per square inch. 

Column 20 had the force applied 2)^ inches out of centre at 
one end, and column 35, 1.9 inches. These tests were made in 
order to observe the effect of eccentricity in the application of 
loads. They show a marked decrease in ultimate resistance. 



Art. 57.] 



LANZA'S EXPERIMENTS. 



481 



TABLE I. 

Timber Mill Columns. 



15 
16 



17 
18 

20 



23 



26 

27 
28 
29 
30 
31 
32 
33 
34 
35 



Round 



Cylindrica 



Square 

Cylindrical 



Round 

Cylindrical. 



Cylindrical 
Square 



Round 



Round 



Cylindric 











SECT. AREA IN 


LENGTH, 
FEET. 


DIA. 


IN INCHES AT 


SQ. INS. AT 












Top. 


Mid. 


BaSe. 


Top. 


Base. 



12.01 
12.02 

12. 01 

12.02 

12. 00 

11.92 

II 98 

2.01 

2.00 

2.01 

2. 00 

12.44 

12.57 

11.93 



1399 
2.00 



11.92 
1.98 

"•93 

12.94 

12.85 

2.00 

2.01 



12.01 
12.01 
12.01 
12.01 
12.00 

6-33 
2. 00 
2 00 

1.98 
"•93 




Yelloiv pine : partially seasoned. 



9^31 


10.65 


10.55 


6589 


85.22 


8.30 


9.71 


10.07 


51.84 


77^36 


7-54 


8.88 


8.99 


42.38 


61.21 


6.40 


7.80 


779 


29.98 


45-47 


10.45 




10.45 


8375 




8.96 




8.96 


61.19 




7.70 




7.70 


44.72 




10.46 




10.46 


85.92 




9.98 




9.98 


78.23 




8.91 




891 


62.35 





7^79 




7-79 


47.66 




8-43 




0.40 


68.80 




8.30 




8.30 


63.10 




9.92 




9.92 


7545 





Yello7u pine : air seasoned. 



7.70 
7.70 



7.90 
7.90 



44.56 
44^56 



47.01 



Yelloiv pine ; dock seasoned. 



8.00 




8.00 


48 


76 





7-93 




7.98 


48 


00 




8.08 




8 08 


63 


28 




8^75 




8.92 


76 


04 





10.05 




10.13 


99 


79 




3-98 




9- 02 


8r 


oo 




10.20 





10.07 


102 


71 





White ivood : partially seasoned- 



White oak : partially seasoned. 



9-13 


10.15 


11.01 


6328 


39.01 


8-37 


9.40 


10.23 


52^83 


80.00 


7-55 


8.75 


9^o5 


42.58 


62.14 


6.60 


779 


8.06 


32.02 


48.83 


10.00 






76 70 




10.00 






76.70 




vl 






78-23 




8.18 






52 55 




7.73 






4'^^93 




8.20 






50.92 





4,098 

3,665 
4,719 

4,602 

4,657 
4,086 

4,584 
4,422 

4,705 
4^330 

4-5II 
5,451 
3,804 
3,512 



4,400 
4,892 



4,662 
3,604 

3,477 
3,682 

5-111 
5,951 
5,452 



12.01 


8.46 1 


9.61 


965 


54.02 


70.95 


3,333 


11.97 


6.38 1 


7.40 


7.72 


29.78 


44.62 


2,687 



3,003 
3,786 
3,758 

3,435 
2,478 
2,738 
3, '32 
3,140 
3,303 
1,964 



Flat 



One flat ; one round. 

Flat. 

One flat ; one round. 

One flat ; one round. 

Flat, pintle. 

Flat. 



31 



482 



TIMBER COLUMNS. 



[Art. 57. 



TABLE I. — Continued, 



36 
37 
38 
39 
40 

41 
42 



43 

44 
45 
46 

47 
48 

49 
50 
51 
52 
53 



54 

55 
56 



Round 


12.08 




12.08 




12.11 




12.05 




11.72 




12.01 




12.07 



Cylindrical . 



Round . . 
Cylindrical. 

Round . . 



Cylindrical . 



LENGTH, 
FEET. 



DIA. IN INCHES AT 



Top. Mid. Base 



SECT. AREA IN 
SQ. INS. AT 



Top. Base 



White oak : used in mill 6% years. 



5-85 
5-85 




6.84 
6.85 


23-89 
24.05 


33-76 
34 -02 


5-87 




6.70 


23.92 


32.12 


6.02 
6.10 




6.75 
6.83 


25-47 
26.08 


32.79 
33-50 


5-97 
5-75 




6.74 
6.88 


25.09 
22.98 


32.78 
34-19 



White oak ; used in will 2$ years. 



-87 


10.56 






84.74 




.00 


10.54 






84.83 




.89 


10.54 







84.40 





.80 


10.50 






83-75 




.92 


10.20 






79-^7 




.89 


10.80 






82.68 




.67 


9-25 




9 50 


64.36 


68.04 


85 


9-55 






68.65 




■65 


9.40 






66.56 




.90 


9-35 







65.82 





-51 


598 




7.20 


26.03 


38.66 



4,604 
6,029 

4,680 
2,945 

3i45i 
4,225 
3)264 



4,602 

4951 
4,266 
3,881 
4.674 
4,838 
3,434 
4,618 

3,981 
3.266 
6,147 



White oak / thoroughly seasoned; i year old. 



12.00 

11.12 

2.00 



7-74 






45.04 




10.95 







91.16 




10.91 






95-48 





3,219 
1,865 

4,450 



Flat. 



Flat, pintle. 
Flat. 



Pintle ends. 
Flat. 



Flat. 

One flat ; one round. 

Flat. 



Although the ends of Nos. 39 and 44-51 were flat, they 
were not parallel. 

All the columns indicated by "Round" were tapered, and 
they almost invariably gave way by the crushing of the fibres 
at the small end. In all such columns the ultimate resistance 
IS per square inch of the small end. 

Some of the square columns had their corners slightly 
beveled. 

In his report to the Boston Manufacturers* Mutual Fire Ins. 
Co., Prof. Lanza says: "The immediate location of the fract- 
ure was generally determined by knots;" . . . but states 



Art. 57.] 



LA ID LEY'S EXPERIMENTS. 



483 



that, whether knotty or straight grained, failure took place in 
the tapered columns at the small ends. Tapering a column, 
therefore, to the extent shown in these cases, is a source of 
weakness. 



TABLE II. 

Yellow Pine. 





LENGTH, 




SECTION DIMENSIONS, 


ULTIMATE RESIST- 


NO. 




FORM OF SECTION. 








INCHES. 




INCHES. 


ANCE PER SQ. IN. 










Lbs. 


I 


20.4 


Circular. 


10.2 Diam. 


6,676 "^ 





2 


119-95 


Square. 


II X II 


6,230 


Td 


3 


119.90 


( i 


II X II 


6,552 


m 

G 


4 


20.0 


<< 


10.4 X 10.4 


7,936 





5 


16.0 


<( 


8x8 


8,165 




6 


8.0 


<< 


4x4 


7,394 


c« 


7 


3-0 


<< 


1.5 X 1.5 


5,533 


13 


8 


6.0 


< t 


3x3 


8,644 


^^S 


9 


6.0 


<( 


3x3 


8,133 




10 


3.0 


i( 


1.5 X 1.5 


8,389 


_G 


II 


3-0 


<< 


1.5 X I 


5 


8,302 


bJO 


12 


30 


(( 


1.5 X I 


5 


6,355 


13 


14.0 


( < 


4.6 X 4 


6 


9,947 


.^ 
.^ 


14 


17.2 


(( 


4.6 X 4 


6 


10,250 




15 


19. 1 


<( 


5-3 X 5 


3 


7,820 J 


4-> 

C/2 


16 


180.0 


Rectangular. 


16 X 13 


65 


3,070 


17 


180.0 


< ( 


16.2 X 7 





2,795 


18 


180.0 


« ( 


17 X 8 


75 


3,180 



N.os. 13, 14 and 15 were pine of very slow growth. 
Nos. 16, 17 and 18 were very green and wet. 

Tables II. and III. contain the results of Col. Laidley's 
tests, some of which belong to short blocks. These tests were 
made during 1881, and a detailed account of them is given in 
"Ex. Doc. No. 12, 47th Congress, 1st Session." 

These experiments give some very important deductions. 

In the first place, within the limits of the ratio of length to 
diameter, or shortest side of rectangular section, appearing in 
these tests, the ultimate resistance is essentially independent 



484 



TIMBER COLUMNS. 



[Art. 57. 



TABLE III. 

Spruce, thoroiighly seasojied. 





LENGTH, 




SECTION DIMENSIONS, 


ULTIMATE RESIST- 


NO, 




FORM OF SECTION. 








INCHES. 




INCHES. 


ANCE PER SQ. IN. 










Lbs. 


I 


24 


Rectangular. 


5-4 X 5-4 


4,946 


2 


24 




5 


4 X 5 


4 


4,811 


3 


36 




5 


4x5 


4 


4,874 


4 


36 




5 


4 X 5 


4 


4,500 


5 


60 




5 


4x6 


4 


4,451 


6 


60 




5 


4x6 


4 


4,943 


7 


120 




5 


4 X 5 


4 


3,967 


8 


120 




5 


4 X 5 


4 


4,908 


9 


60 




5 


4x5 


4 


5,275 


10 


30 




5 


4 X 5 


4 


5,372 


II 


15 




5 


4 X 5 


4 


5,754 


12 


121. 2 


Circular. 


12 


4 Diam. 


4,681 



of the length. This is the result of the action of causes noticed 
in the consideration of wrought-iron columns composed of C's. 
The ultimate resistance of any such column, therefore, is to be 
obtained by multiplying the area of its cross section by the 
ultimate resistance, per square inch, of short blocks. 

In Prof. Lanza's experiments, the greatest ratio of length 
to radius of gyration was about 86. Below this value the gen- 
eral conclusion just given may be expected to hold, but prob- 
ably not much above it. 

In Col. Laidley's tests the greatest value of the same ratio 
was about 90 (No. 17 of Table II.), at which there seemed to 
be a little decrease in ultimate resistance. 

Again, it is to be observed that Prof. Lanza's results are 
much less than those of Col. Laidley for the same timber. The 
columns of the former were of ordinary merchant material, 
with the usual accompaniment of knots, weak spots, crooked 
grain, etc., while the latter experimented with fine, straight- 
grained timber. 



Art. 57]. a SHALER SMITH'S FORMULA, 485 

The slow growth specimens (13, 14 and 15, of Table II.), 
gave much the highest results, while the wet and unseasoned 
ones (16, 17 and 18) gave the lowest of all. 

Hence, the ultimate resistance of timber columns will de- 
pend upon quality and condition of material, mode of growth, 
degree of seasoning, etc., etc. 

Table II. also shows, what has been observed elsewhere, 
that smaller specimens give higher results than larger ones. 



Formula of C. Shaler Smithy C, E. 

During the winter of 1861-62, Mr. C. Shaler Smith con- 
ducted a series of over 1,200 tests of full size yellow pine 
square and rectangular columns for the Ordnance Dept. of the 
Confederate Government. The results of these tests have 
never been published, but Mr. Smith has kindly furnished the 
writer with the following summary : 

The tests were grouped as follows : 

'' 1st. Green, half-seasoned sticks answering to the specifi- 
cation, * good merchantable lumber.' 

*' 2d. Selected sticks reasonably straight, and air-seasoned 
under cover for two years and over. 

^* 3d. Average sticks cut from lumber which had been in 
open air service for four years and over." 

If / = length of column in inches ; 
d = least side of column section in inches ; 
and / = Ult. Comp. resistance in lbs. per sq. in. ; 

then the formulae found for these three groups were : 

For No. I : / = — 5.400 



r+ ' ^'" 



250 d'- 



4^6 TIMBER COLUMNS. [Art. 57. 



For No. 2 : / = 



8,200 



I H — 

300 d^ 



T- ' AT X 5,000 

For No. 3 : / = ^^^ ~ . 

^ 250 d"" 

But in order to provide against ordinary deterioration while 
in use,, as well as the devices of unscrupulous builders, Mr. 
Smith recommends the formula for group No. 3 as the proper 
one for general application. He also recommends that the 

factor of safety shall be a - until 25 diameters are reached, 

and five thenceforward up to 60 diameters. This last limit he 
regards as the extreme for good practice. 

Mr. Trautwine computed his tables from tests of group 

No. 3. 



CHAPTER VIII. 

Shearing and Torsion. 

Art. 58. — Coefficient of Elasticity. 

It has already been shown in some of the Articles of the 
first portion of this book, on shearing and torsion, that the co- 
efficients of elasticity for those two stresses are the same ; and, 
indeed, that those two stresses are identical in character. The 
coefficients of elasticity, given in this Article, are then derived 
chiefly from experiments in torsion. 

In his " Legons de M^canique Pratique," 1853, Gen. Arthur 
Morin gives the following resume of the results of experiments 
up to that time, in which G is the coefficient of elasticity, for 
shearing, in pounds per square inch. 

MATERIAL. C, IbS. 

Soft wrought iron 8,571,000 

Iron bars 9,523,000 

German steel 8,571,000 

Fine cast steel 14,300,000 

Cast iron 2,857,000 

Copper 6,237,000 

Bronze 1,523,000 

Oak 571,400 

Pine.. 618,600 

The above value for cast iron must, however, be much too 
small, as will presently be seen. 

In '* Der Civilingenieur," Heft 2, 1881, the results of some 
very interesting and important experiments on cast-iron rods 
or prisms of various cross sections, by Prof. Bauschinger, are 



488 SHEARING AND TORSION. [Art. 58. 

given in full detail. The rods or prisms were about 40 inches 
long, and were subjected to torsion, while the twisting of two 
sections about 20 inches apart, in reference to each other, was 
carefully observed. The results for four different cross sections 
will be given — i. e., circular, square, elliptical (the greater axis 
was twice the less) and rectangular (the greater side was twice 
the less). In each case the area of cross section was about 
7.75 square inches. The angle a is the angle of torsion — i. ^., 
the angle twisted or turned through by a longitudinal fibre, 
whose length is unity, and which is at unit's distance from the 
axis of the bar. * 



G. 



Circular . 



j 0.007 degree 7,466,000 lbs. per sq. in. 



.07 " 6,157,000 

ElliDtical 30-009 " 7,437.000 

-^^^^P^^^^^ ] 0.076 " 6,228,000 

Sauare i 0.008 " 7,039-000 

^"l"^"^^ 1 0.073 " 5,987,000 

01 " 6,996,000 

08 " 5,716,000 



Rectangular. 



lo: 



The formula by which G is computed, when the torsional 
moment and angle a are given, is the following : 

G = ^ .cli- (I) 



in which M is the twisting moment ; A the area of the cross 
section ; Ip the polar moment of inertia of that cross section ; 
and c a coefficient which has the following values : 

471^ — 39.48 for circle and ellipse ; 
42.70 " square ; 
42.00 '* rectangle ; 

as was shown in Art. 10. 

Bauschinger's experiments show that the coefficient of 



Art. 58.] COEFFICIEXTS OF ELASTICITY. 489 

shearing elasticity for cast iron may be taken from 6,000,000 to 
7,000,000 pounds per square inch ; also, that it varies for differ- 
ent ratios between stress and strain. 

It has been shown in Art. 4, that if E is the coefficient of 
elasticity for direct stress, and r the ratio between direct and 
lateral strains, for tension and compression, that G may have 
the following value : 

^ = J(IT^ ^^) 

Prof. Bauschinger, in the experiments just mentioned, 
measured the direct strain for a length of about 4.00 inches, 
and the accompanying lateral strain along the greater axis of 
the elliptical and rectangular cross sections, and thus deter- 
mined the ratio r between the direct and lateral strains per 
unit, in each direction. The following were the results : 

CoDipressioii, 

SECTION. r. G. 

Circular 0.22 6,541,000 lbs. per sq. in. 

Elliptical 0.23 6,541,000 " " " " 

•Square 0.24 6,442,000 " " " " 

Rectangular 0.24 6,499,000 " " '' " 

Tension. 

Circular "0. 23 6, 570,000 lbs. per sq. in. 

Elliptical 0.21 6,811,000 " 

Square 0.26 6,399,000 " 

Rectangular 0.22 6,527,000 " 



( ( ( ( ( < 



The values of E are not reproduced, but they can be calcu- 
lated indirectly from Eq. (2) if desired. 

It is seen that the values of G, as determined by the differ- 
ent methods, agree in a veiy satisfactory manner, and thus fur- 
nish experimental confirmation of the fundamental equations 
of the mathematical theory of elasticity in solid bodies. 



490 



SHEARING AND TORSION. 



[Art. 59. 



The fact that G is essentially the same for all sections is also 
strongly confirmatory of the theory of torsion, in particular. 

These experiments show that, for cast iron, the lateral strains 
are a little less than one quarter of the direct strains. If rwere 
one quarter, then 6^ = -|^;or^=:|G^. 

In the ''Journal of the Franklin Institute," for 1873, Prof. 
Thurston gives the following values of G, as determined from 
experiments with his torsion machine. 



White Pine G — 

Yellow Pine, sap G =■ 

Yellow Pine, heart 6^ = 

Spruce 6* = 

Ash 6^ = 

Black Walnut G = 

Red Cedar Cr = 

Spanish Mahogany G ^ 

Oak G = 

Hickory 6^ = 

Locust 6^ = 

Chestnut 6^ = 



220,000 pounds per sq. in 

495,000 " 
495,000 

211,000 " " 

410,000 " " 

582,000 " " 

890,000 " " 

660,000 " " 

570,000 " " 

910,000 " " 

1,225,000 " " 

355,000 '' 



The specimens were small ones, and the timber was sea- 
soned. 



Art. 59. — Ultimate Resistance. 

Before considering the ultimate shearing resistance of spe- 
cial materials it will be well to notice the .two different methods 
in which a piece may be ruptured by shearing. 

If the dimensions of the piece in which the shearing force 
or stress acts are very small, i.e., if the piece is very thin, the 
case is said to be that of '' simultaneous " shearing. If the 
piece is thick, so that those portions near the jaws of the shear 
begin to be separated before those at some distance from it, 
the case is said to be that of " shearing in detail." In the lat- 
ter case failure extends gradually, and in the former takes place 
simultaneously over the surface of separation. Other things 



Art. 59.] ULTIMATE RESISTANCE. 49 1 

being the same, the latter case (shearing in detail), will give 
the least ultimate shearing resistance per unit of the whole 
surface. 

In reality no plate used by the engineer is so thin that the 
shearing is absolutely simultaneous, though in many cases it 
may be essentially so. 



Wrought Iron, 

The following averages (each result being an average of 
six tests), are from Chief Engineer Shock's experiments, in 
1868, on ordinary commercial " rounds " (*' Steam Boilers," by 
William H. Shock, Chief Engineer, U. S. N.), in which 5 is the 
ultimate shearing resistance in pounds per square inch : 



DIAM. OF ROUND. SINGLE SHEAR. DOUBLE SHEAR. 

0.5 inch 44,150 lbs 41,090 lbs. 

0.625 inch 39,250 lbs '. . 38,670 lbs. 

0.75 inch 39,550 lbs 39,770 lbs. 

0.875 inch 41,500 lbs 37,890 lbs. 

1. 00 inch 40,700 lbs 37,650 lbs. 



Although these figures show some irregularities, the general 
result is unmistakable, and shows a decrease of 5 with an in- 
crease of diameter. 

The results of experiments at Bristol, England, by Mr. 
Jones (*'Proc. Inst. Mech. Engrs.," 1858), on punching plate 
iron, are as follows : 

THICKNESS OF PLATE. DIAM. OF HOLE. S. 

0.437 inch 0.250 54,700 lbs. per sq. in. 

0.625 " 0.500 60,900 " " " " 

0.625 •' 0.750 52,900 " " " " 

0.875 " 0.875 51,700 " " " " 

1. 000 " 1. 000 55,100 " '* " " 



492 SHEARING AND TORSION. [Art. 59. 

Mr. C. Little found the following for English '' hammered 
scrap bars and rolled iron," with parallel cutters or shears : 

AREA CUT. DIRECTION. S. 

0.50 X 3.00 ins Flat 49,950 lbs. per sq. in. 

0.50 X 3.00 ins Edge 5i,750 

1. 00 X 3.00 ins Flat 5i,750 

1. 00 X 3.00 ins Edge 50,850 

1. 00 X 3.02 ins Flat 44,350 

1. 00 X 3.02 ins Edge 46,150 

1.80 X 5.00 ins Edge 46,150 

In these experiments the edges of the shears were always 
parallel to each other, thus tending to produce simultaneous 
shearing. In ordinary workshop practice, however, the jaws of 
the shears make a constant angle with each other, thus shear- 
ing successive portions of the material as the jaws approach, 
whatever may be the dimensions of the piece, and conse- 
quently always producing shearing in detail. In the experi- 
ments (by the same authority, /. e., Mr. C. Little, " Proc. Inst. 
Mech. Engrs.," 1858) from which the following results were 
deduced, the angle between the jaws of the shears was an incli- 
nation of I in 8 : 

BARS. FLATWAYS. EDGEWAYS. 

3 X 1 . 5 ins S — 40,800 45,000 lbs, per sq. in. 

4.5 X 1.375 ins 6" = 32,000 40,100 " " " " 

3.0 X 1. 00 ins .5=35,200 47.300 " " " " 

5.25 X 1.75 ins 5^=37,400 50,600 " " " " 

6.00x1.50 ins 6" = 33,600 41,200 " " " " 

As was to be expected, the '^Edgeways'' results are much 
the largest, as with that position of the bar the shearing ap- 
proached more nearly the simultaneous condition. These 
results show that it is much more economical to shear a bar 
flatways than edgeways. 

Mr. Edwin Clark (''On the Tubular Bridges") found the 
resistance of ^-inch rivet iron, in single and double shear, to 



Art. 59.] ULTIMATE RESISTANCE. 493 

vary from 49,500 to 54,100 pounds per square inch. The ten- 
sile resistance of the same iron was about 53,800 pounds per 
square inch. 

Reviewing all these results, the ultimate shearing resistance 
of wrought iron may safely be taken at 0.8 of its tensile resist- 
ance, as stated by Mr. D. K. Clark. 



Cast Iron. 

Very few experiments on the resistance of cast iron to 
shearing have been made, as this metal is seldom or never used 
to resist such a stress. 

Mr. Bindon B. Stoney ("Theory of Strains in Girders and 
Similar Structures," p. 357 of 2d Edit.) has found, by experi- 
ment, that the ultimate shearing resistance of the cast iron 
with which he experimented varied from about 17,900 to 
20,200 pounds per square inch. He concluded that the shear- 
ing and tensile resistances might be taken the same. 



Steel. 

In 1866 Mr. Kirkaldy tested the ultimate shearing resist- 
ance of hematite bar steel made by the Barrow Hematite Steel 
Company, of Great Britain, and found the mean of four experi- 
ments with hammered cast steel rounds 1.25 inches in di- 
ameter, to be about 

56,500.00 pounds per square inch. 

The tensile resistance of the same steel was found to be 
about four-thirds as much. 

The same experimenter investigated the ultimate shearing 
resistance of four grades of Fagersta steel, and the following 
results are taken from " Experimental Enquiry into the Me- 



d. 




0. 


.19 


inch. 


o. 


• 25 


( ( 


o, 


.28 
■ 32 


(( 





( < 



494 SHEARING. [Art. 59. 

chanical Properties of Fagersta Steel," by David Kirkaldy, 
1873. The test-piece, in each case, was turned from a 2-inch 
square bar, to a diameter of 1.128 inches, and each result is a 
mean of three experiments. S is the ultimate resistance to 
shearing, in pounds per square inch ; r is the ratio of ultimate 
shearing over ultimate tensile resistance of the same steel ; 
while "^" is the detrusion or relative movement of one part of 
the specimen in respect to the other at the instant of separa- 
tion over the entire surface. 

MARK. S. r. 

1.2 61,400.00 lbs 0-73 

0.9 79»740-oo " 0.75 

0.6 71,650.00 " 0.70 

0.3 45,410.00 " 0.74 

As is evident, the lower ^^ Mark'' numbers belong to the 
softer steels. 

In each case two surfaces were sheared, as the " round " 
was a pin for three links, two of which pulled one way, and 
one the other. 

All of Mr. Kirkaldy's experiments seem to show that the 
ultimate shearing resistance of steel is about three-quarters the 
tensile. 

Table I. contains the results of the experiments of Prof. A. 
B. W. Kennedy, as given in "Engineering" for May 6, 1881. 
The tensile resistance of the same steel was given in the chap- 
ter on " Tension." 

The specimens were round and of mild rivet steel. The 
ratio of the ultimate resistance to shearing over that to tension 
varied from 0.80 to 0.89. 

In the ** Journal of the Franklin Institute," for March, 1881, 
Charles B. Dudley, Ph.D., gives the results of 192 tests of rail 
steel, the specimens, 0.625 inch round, having been taken from 
rails which had been subjected to service for considerable pe- 
riods of time on the Penn. R. R. The tests were made by 



Art. 59.] 



STEEL AND COPPER. 



495 



TABLE I. 

Rivet Steel. 





ULTIMATE RESIST. IN 




RATIO OF ULT. SHEAR 


DIAMETER IN INCHES. 




MEAN. 






LBS. PER SQ. IN. 




OVER ULT. TENSION. 


I.OO 


54,110] 








I.OO 


54,930 








I.OO 
I.OO 


55,240 
52,830 


\ 


54,550 


0.89 


I.OO 


56,660 








I.OO 

0.62 


53.530J 
60, 260 ~ 








0.62 
0.62 


59,400 
59.600 


- 


59,640 


0.87 


0.62 


59,220 








0.62 
62 


59, 740 J 
53,670] 








0.62 
0.62 


51,290 
52,670 


- 


52,450 


0.80 


0.62 


53,000 








0.62 


.51,620^ 









Mr. J. W. Cloud, engineer of tests for the Penn. R. R. Co. 
The following is a summary of the results : 

{63,560.00 pounds per sq. in. i^greatesf). 
59,880.00 pounds per sq. in. {inean). 
53,380.00 pounds per sq. in. (least). 

The percentages of carbon and ultimate tensile resistances 
are given in Table IV. of Art. 34. By reference to that table 
it will be observed that vS is not far from three-fourths the 
tensile resistance. 

Copper. 

From some English experiments, Mr. Bindon B. Stoney 
concluded that the ultimate shearing resistance of copper was 
about two-thirds of that of wrought iron. 



496 



SHEARING. 



[Art. 59, 



Timber, 

In treating the shearing resistance of timber, it is very nec- 
essary to consider whether the shearing takes place along the 
fibres, or in a direction normal to them. 

TABLE I. 

Along Fibres. 





S. 


KIND OF WOOD. 


GREATEST. 


MEAN. 


LEAST. 


Georgia Pine 


934 

530 
1,389 

1,474 

647 
410 


843 
482 
1,165 
1,250 
542 
369 


713 

433 

970 

I 076 




White Pine 


Locust 


White Oak 


Spruce 

Hemlock 


463 
322 





TABLE IL 

Across Fibres. 



KIND OF WOOD. 


S. 


KIND OF WOOD. 


6-. 


Ash 


6,280 
5,223 

5,595 
1,372 to 1,519 
3.410 
2,945 
1,535 
6,510 

7,750 

5,890 

2,750 

6,045 to 7,285 


Locust 


7.176 
6,355 
4,425 
8,480 
2,480 
4,340 
5.735 
5,053 
4,418 
3,255 
4,728 
2,830 


Beech 


Maple 


Birch 


Oak, white 


Cedar, white 


Oak, live 


Cedar, Central Amer. . . . 


Pine, white 


Cherry 


Pine, yellow, northern. . 
Pine, yellow, southern.. 
Pine, yel. , very resinous. 
Poplar 


Chestnut 


Dogwood 


Ebony* 


Gum 


Spruce 


Hemlock 


Walnut, black 


Hickory 


Walnut, common 







Art. 59.] 



TIMBER. 



497 



Table I. contains the results of experiments on the shearing 
of small specimens a/on^ the fibres, by the late Mr. R. G. Hat- 
field (''Transverse Strains," 1877). 5 is the ultimate shearing 
resistance in pounds per square inch. There were about nine 
experiments for each kind of timber. 

Table II. contains the results of experiments by Mr. John 
C. Trautwine on round specimens 0.625 inch in diameter, and 
across the fibres ("Journal of the Franklin Institute," Feb. 

TABLE III. 

Along Fibres. 



3 
4 

5 

6 

7 
8 

9 

10 
II 
12 
13 
14 
15 
16 

17 
18 

^9 
20 
21 
22 
23 
24 
25 
26 

27 
28 

29 
30 
31 
32 
3i 
34 
35 



KIND OF MATERIAL. 



Oregon pine 

Oregon pine 

Oregon maple 

Oregon maple 

California laurel.. 

Ava Mexicana 

Ava Mexicana 

Oregon ash 

Oregon ash . . 

Mexican white mahogany 

Mexican cedar 

Mexican cedar 

Mexican mahogany 

Mexican mahogany 

Oregon §pruce 

Oregon spruce 

White pine 

White pine 

Whitewood 

Whitewood 

Whitev/ood 

Yellow pine. . . 

Yellow pine 

Ash 

Ash 

Red oak 

Red oak 

White oak 

White oak 

Yellow birch 

Yellow birch 

White maple 

White maple 

Spruce 

spruce 



SHEARING AREA 

IN SQUARE 

INCHES. 



;.o and 14.0 

10.7 

14.4 

10. g 
: .0 and 14.2 

14.8 

31 .0 
14.6 

8.2 

:.o and 15. i 

15.0 
9.8 

15.0 

II. I 
!,9 and 34.5 

5 5 
!.o and 16.0 
).o " 24.0 



32 



13- 
17- 
16. 
16. 
16. 
16. 
16. 
15- 

25. 

15 
15 
16 



25.4 

" 24.4 

" 16.0 

" 23.9 

" 24.0 

" 24.0 

" 24.0 

" 17 o 

" 25.6 

16.0 

9 and 24.0 

8 '' 23.8 

o " 24.0 



ULT. SHEAR IN 

POUNDS PER SQ. 

INCH. 



442 and 1,096 

820 

436 
1,028 
549 and 1,204 

346 

700 

443 

1,126 

438 and 1,000 

423 
814 
566 

i«333 
261 and 



315 
381 and 

324 " 

127 " 

328 " 

322 '' 

317 '' 
286 ' 

592 ' 
458 ' 
743 ' 
726 ' 
803 ' 
752 ' 
563 ' 
612 ' 

647 
399 ^nd 
235 " 
316 " 



356 

423 

352 
370 
481 

385 
399 
409 
600 
700 
745 
999 
966 
846 

815 
672 

537 
347 
374 



DIRECTIONS TO RINGS 
OF GROWTH. 



Perpendicular (2 exps.) 
Oblique. 
Perpendicular. 
Oblique. 

Oblique (2 exps.) 
Perpendicular. 
Parallel. 
Parallel. 
Perpendicular. 
Oblique (2 exps.) 
Perpendicular. 
Parallel. 
Parallel. 
Perpendicular. 
Parallel (2 exps ) 
Perpendicular. 
Perpendicular (2 exps.) 
Parallel (2 exps ) 
Oblique (2 exps ) 
Parallel (2 exps.) 
Perpendicular (2 exps.) 
Oblique (2 exps.) 
Perpendicular (2 exps.) 
Parallel (2 exps.) 
Perpendicular (2 exps.) 
Perpendicular (2 exps.) 
Parallel (2 exps.) 
Parallel (2 exps ) 
Perpendicular (2 exps.) 
Oblique (2 exps.) 
Perpendicular (2 exps.) 
Oblique. 

Perpendicular (2 exps.) 
Parallel (2 exps.) 
Perpendicular (2 exps.) 



32 



49^ TORSION. [Art. 60. 

1880). As before, vS is the ultimate shearing resistance in 
pounds per square inch. 

Table III. has been condensed from the results of Col. 
Laidley's tests at the Watertown Arsenal (Ex. Doc. No. 12, 
47th Congress, ist Session). Usually, two such results have 
been selected as will give a correct idea of the resistance. In 
all cases except Nos. 19, 20, 23 and 33, the smaller resistance 
belongs to the larger shearing surface. In No. 33 the smaller 
resistance belongs to an unsatisfactory experiment. 



Art. 60. — Torsion. 

Coefficients of Elasticity. 

The coefficients of elasticity for torsion or shearing have 
been given in Art. 58, and need not be repeated here. 

Ultimate Resista?7ce and Elastic Limit. 

Wrought Iron. 

In 1866 Mr. Kirkaldy tested four hammered Swedish iron 
bars turned to a diameter of 1.5 inches for a length of seven 
diameters. The average ultimate moment of torsion was pro- 
duced by a weight of 2,677 pounds with a leverage of 12 
inches; hence, in Eq. (83) of Art. 10 ; J/ = 2,677 X 12 = 
32,124. Putting 2r^ — d — 1.5 inches in that equation, there 
will result : 

M ' 1 

7;^ = 5.1 -— = 48,540 pounds per square men. 

d"^ 

This is the greatest intensity of torsional shear in the sec- 
tion. 



Art. 60.] WROUGHT IRON. 499 

If T^ be taken at 48,000 the diameter of a wrought-iron 
shaft required to resist an ultimate moment M^ will be : 

d = 0.047 ^/~M (i) 

If the working moment be taken at one-eighth the ulti- 
mate, then the diameter required will be : 



d = 0.047 V 8 M^ = 0.094 ^M, ... (2) 

in which M^ is the working moment. 

If H is the number of horse powers per minute to be trans- 
mitted by the shafting, and n the number of revolutions which 
it is to make : 

12 X 33.000 _H 

27t n ^ ' 

Putting this value in Eq. (2) : 



</=3.742^/J • • (4) 



This value of <a^ will be much too small in the case of long 
shafting required in the distribution of power, in consequence 
of the bending caused by the belting. 

The mean torsional moment at the elastic limit, in Mr. 
Kirkaldy's four experiments, was about 0.4 the ultimate. 

In 1846 Major Wade ('^ Experiments on Metals for Can- 
non ") tested three wrought-iron circular cylinders about 1.9 
inches in diameter and 15 inches long, with the following re- 
sults : 



500 



TORSION. 



Art. 60.] 



T 



_ ^aM _ 



^3 



= 28,325 lbs. per sq. in. 



= 27,525 '' 
= 27,800 " 






83,650 



Mean = 27,900 (nearly). 
If the mean be taken at 28,000 : 

d = 0.056 y'M . 



(5) 



It is seen that Major Wade found T^ much less than Kirk- 
aldy's value for Swedish iron, and d in Eq. (5) is correspond- 
ingly greater. If H and n carry the same signification as 
before, and if 8 is the safety factor : 



d=4'49 \'/^ 



(6) 



In all these results, the moments are supposed to be given 
in inch-pounds^ and the resulting values of d are consequently 
in inches. 

Cast Iron. 

Major Wade also made tests on circular cylinders of cast 
iron about 1.9 inches in diameter and 15 inches long. 

If d is the diameter = 2r^ in Eq. (83) of Art. 10, he found 
the following results with the grades of iron shown : 

2d fusion Tm = 3i,500 pounds per square inch. 

3d fusion 7^^ = 44,775 " 

2d and 3d fusion T,n — 49^735 

2d fusion T,n = 40, 020 ' ' 

3d fusion T,n — 53,380 '' 

2d fusion l\t = 49, 526 " 

3d fusion T,n = 46.230 " 

Mean = 45,000 (nearly). 



Art. 60.] CAST IRON, 501 

Hence the diameter in inches, for the ultimate moment M 
in inch-pounds is : 



'^=A/i;^oo^=-°48Ay'^ .... (7) 



These values of T^ are very high, because the iron with 
which Major Wade experimented was evidently of a special 
character and extraordinarily strong. 

The same experimenter tested some square sections, for 
which, by Eq. (73) of Art. 10 : 

M 
T = ^ — \ {b ~ side of square) .... (8) 



The following are from Major Wade's results : 

b = 1.00 inches ; M = 8,750 inch-pounds ; T^n = 43,75© pounds. 
d = 1.42 inches ; M = 23,000 inch-pounds ; Tm = 40,210 pounds. 
d = 1.75 inches ; M = 54,000 inch-pounds ; T^ = 50,370 pounds. 

The mean of these results is : 7" = 44,800 (nearly). 
Hence for the ultimate moment in inch-pounds : 



It is to be observed that, according to these experiments, 
T^ is the same for circular and square sections ; a result very 
different from that of Prof. Bauschinger's experiments, as will 
presently be seen. 

Four of Major Wade's experiments on hollow circular cyl- 
inders are next to be given. 

Since T^^ — o, in Eq. (78) of Art. 10, the resisting moment 



502 TORSION. [Art. 60. 



of such a cylinder, if d is the external and d^ the internal di- 
ameter, will be : 

M = ^^^ ^-^^ = -^ U^ - dA . . . (10) 

5.1 5.1^ ^ ^ 

^ S'ldM . . 

••• ^- = ^^7^4 00 

For the first case : 

d = 3.25 ins. ; d^^ = 2.61 ins.; M = 95,000 in.-lbs. 
Substituting in Eq. (11) : 

T^ = 24,170 lbs. per sq. in. (nearly). 

For the second case : 

d = 2.21 ins. ; d^ = 1.54 ins. ; M = 49,500 in.-lbs. 
Substituting in Eq. (ii) : 

^m = 30,610 lbs. per sq. in. (nearly). 
For the third case : 

d — 1. 81 ins. ; d^ = 0.91 in. ; M = 37,250 in.-lbs. 
Substituting in Eq. (11) : 

^m = 34,220 lbs. per sq. in. (nearly). 
For the fourth case : 

d = 1.30 ins. ; d^ = 0.65 in. ; M = 13,000 in.-lbs. 



Art. 60.] CAST IRON. 503 

Substituting in Eq. (11) : 

T^ — 32,180 lbs. per sq. in. (nearly). 

These results indicate that T^ decreases as the thickness of 
the wall of the hollow cylinder decreases and as the exterior 
diameter increases. 

Professor Bauschinger (Der Civilingenieur, 1881, heft 2) 
tested cylinders about 40 inches long, and with the following 
cross sections and approximate dimensions : 

Circle Diameter = 3.25 inches. 

T-,„. T>>- . ] 2.30 inches. 

E"'P^^ Diameters = \ ^ ^^ -^^^^^^^^ 

a c-j i 3-00 inches. 

S^""^^ Sides = 1 3.00 inches. 

Rectangle Sides = ] ^ " ^-^ !"^!j^s- 

° ( 4- 10 inches. 

Rectangle Sides = Ji-^^ inches. 

^ ( 4- 10 inches^ 

The ultimate twisting moments substituted in Eqs. (83), 
(41)^ (73), (75), and {77) of Art. 10, give : 

For Circle Tm = 27,730 pounds per square inch. 

For Ellipse T^ = 36,120 pounds per square inch. 

For Square T^ — 37ii6o pounds per square inch. 

For Rectangles (sides 2 to i). . T,n = 36,370 pounds per square inch. 

For Rectangles (sides 4 to i). . T„t = 37,090 pounds per square inch. 

These experiments give T,^ considerably less value for the 
circular cross section than for the others. 

The U. S. Board, however, found the following values for 
four cast-iron cylinders one inch long and 0.565 inch in diam- 
eter: 

^m = 35^980; 34>iio; 34^280, and 33,770 lbs. per sq. in. 

£/as. Lhn. = 60; 55 ; 64, and 62 per cent, of 7"^, respect- 
ively. 



504 TORSION. [Art. 6o. 

Steel. 

In connection with the torsional resistance of steel, tests of 
circular cylinders only are to be considered. Those to first re- 
ceive attention were made by Mr. Kirkaldy on English steel, 
in 1866-1870, and the results have been deduced from his data. 

As the sections are all circular, Eq. (83) of Art. 10 is the 
only one needed: 

T. = '-^ (-) 

In this equation T^ Is the greatest intensity of torsional 
shear, in any section, in pounds per square inch; "<^" the di- 
ameter of the shaft or cylinder in inches ; and M the twisting 
moment in incJi-pounds. 

In all the following experiments the lever arm of the twist- 
ing couple was 12 inches; hence, if P is the twisting force, 
M = i2Py and Eq. (12) becomes 

^ 61. 2P 

T^ = -j^ (13) 

The mean of four experiments with Bessemer steel gave 
for the ultimate resistance 

P = 2,307 lbs., with d = 1.25 inches ; 

.*. T^ = 72,298 lbs. per sq. in (14) 

The length was 10 inches. 

The mean of some results with Krupp's cast steel in speci- 
mens 1.25 inches in diameter, and 2.5 inches for torsion length, 
gave: 

P = 2,867 lbs. .-. T^ = 89,847 lbs. . . (15) 



Art. 60.] 



STEEL. 



505 



The following set of results were obtained from 2-inch 
square bars turned down to 1.382 inches in diameter for a 
length of 1 1 inches, and gives the means of the number of tests 
indicated. 



SPECIMENS. 

5 Hammered tires, 
5 " axles, 

4 " rails, 

4 Rolled tires, axles 

and rails. 

5 Hammered tires, 
4 " axles, 
I " rail, 

I Rolled rail. 



■ ^1^ 

l/l w 

03 






P (ultimate). 


T^ (ultimate) 


. 3,450 lbs. . 


. . 80,006 lbs. 


. 3.293 " .• 


.. 76,365 " . 


. 3,248 " .. 


•• 75,321 " . 


. 3,226 " .. 


.. 74,811 " . 


. 3.562 " .. 


. . 82,603 " • 


. 3,786 " .. 


.. 87,797 " . 


. 4,054 " .• 


. . 94,012 " . 


. 3,002 " . 


.. 69,616 " . 



ELASTIC 
STRAIN. 

0.014 

O.OII 

0.012 

0.008 >• 

0.014 

0.013 

0.016 

0.012 



• (16) 



The elastic strain is the fraction of a complete turn made 
by the specimen at the elastic limit. 

The mean of the Bessemer steels in (14) and (16) give: 

Mean 7",^ = 75)7^0 lbs. per sq. In. 

Hence, if 3f is the breaking moment of the twisting couple 
in incJi-poiinds^ the following will be the diameter of the shaft 
in inches : 



'^"V^^ = °-°4°7^''^: • • 



• • ('7) 



Or, if n is the safety factor, and M^^ the greatest working 
moment : 

d = o.0407^Jm^ (18) 

The mean of the crucible steel results in (16), with the ex- 
ception of the last, is: 



Mean T^ = 88,140 lbs. 



5o6 TORSION. [Art. 60. 

Hence the diameter (in inches) of the shaft which will just sus- 
tain the breaking moment M, in inch-pounds^ is: 



^=A/i;i^=^-0387^J/ .... (19) 



Or, if n is the safety factor, and M^ the greatest working 
moment : 



d =: o.0387V?zi/, (20) 

In all the preceding experiments the elastic limit varied 
from 40 to 47 per cent, of T.^ (ultimate) as given in (14), (15) 
and (16). 

In 1873 Mr. Kirkaldy made some experiments on specimens 
of Fagersta steel which possessed a length of about 9 inches 
and a diameter of 1.128 inches, the length of the twisting lever 
being still 12 inches. Eq. (13) then gives the following results, 
each being a mean of three tests: 

MARK. P (ultimate). T^^^ (ULTIMATK). STRAIN. 

1.2 2,120 lbs 90,397 lbs 0.29 

0.9 2.336 •' 99.607 " 0.79 

0.6 2,261 " 96,409 " 1.02 

0.3 1,520 " 64,813 " 3.22 

The ^'■strain'" is the number of complete turns made by the 
specimen at the place and instant of rupture. 

The specimens with the higher "mark" numbers were the 
higher steels. 

The elastic limit varied from 46 to 58 per cent, of the ulti- 
mate r,„. 

The diameter of a shaft for any of these grades may readily 
be computed by the use of these values of 7^ in equations 
similar to Eqs. (17) to (20). 



Art. 60.] 



STEEL. 



507 



The following values were determined by the Committee 
on Chemical Research of the U. S. Board, '■' Ex. Doc. 23, 
House of Rep., 46th Congress, 2d Session," with specimens I 
inch long turned to diameters of 0.625 and 0.565 inch, and 
tested in a Thurston machine: 



ELAS. LIMIT IN PER CENT. 

OF t:... 



ULT. ANGLE OF 
TORSION. 



100,990 lbs, per sq. in 34 149° .0 



95,230 
110,260 
115,780 

52,375 
71,420 
88,210 
55,885 

119,040 
75,430 
91,690 
96,450 

109,010 

107,315 
109, 590 



34. 
33. 
42. 

34. 

45. 
39- 
35- 
40. 



.142 .3 

, 68°. 4 

56°. I 
278^.2 
,220°. 8 

99° -5 
.i65\o 
, 84\9 



44 180^7 

39 53' to 113' 

36 48 Mo 84° 

29 61° to 143° 

32 42° to 123' 

32.. 73° to 141° 






Each of the last five results is a mean of eight tests. 

The first portion of these results would possess more value 
if the test specimens had been larger. 

With these values of T^, the diameter of a shaft, with the 
torsion moment Min inch-pounds, becomes: 





Copper, Tin, Zinc, and their Alloys. 

The following values of T^ have been computed by the aid 
of Eq. (12) from data determined by Prof. R. H. Thurston, and 
given by him in the works already cited in connection with 



5o8 



TORSION. 



[Art. 60. 



tension and compression. The test specimens were 0.625 inch 
in diameter, with a torsion length of i.oo inch, and were tested 
in his torsion machine. The ultimate shearing resistances of 
these alloys in torsion are thus seen to vary as widely as their 
tensile resistances. 

TABLE I. 



COMPOSITION. 










ULTIMATE TORSIVE 


ELASTIC LIMIT ; PER 
CENT. OF T„i. 


ULTIMATE TORSION 






ANGLE. 


Cu. 


Sn. 












Pounds. 




Degrees. 


100 


00 


35,910 


35 


I53-0 


100 


00 


28,430 


40 


52 to 154 


00 


100 


3.196 


45 


557-0 


00 


100 


3,297 


33 


691.0 


90 


10 


43,943 


41 


II4-5 


80 


20 


47,671 


62 


16.3 


70 


30 


4,407 


100 


1-5 


62 


38 


1,770 


100 


I.O 


52 


48 


686 


100 


I.O 


39 


61 


5,881 


100 


1.7 


29 


71 


5,257 


TOO 


2.34 


10 


90 


5,761 


63 


131-8 


90 


10 


25,027 


49 


57.2 


90 


10 


31,851 


57 


72.6 



T^fi is in pounds per square inch. 



Table I. relates to alloys of copper and tin, and Table II. 
to alloys of copper and zinc. 

None but specimens with circular sections were tested. 

With the preceding values of T^, the following expression 
for the diameter in inches may be written, if M is given in inch- 
pounds : 



'c^.iM 



T 

"*■ -M 



1. 721 




Art. 60.] 



COPPER-ZINC ALLOYS. 



509 



TABLE 11. 



PERCENTAGE OF 










ULTIMATE TORSIVE 
SHEAR, T^^. 


ELASTIC LIMIT ; PER 
CENT. OF T^. 


ULT. TORSION 


Copper. 


Zinc. 


ANGLE. 






Pounds. 




Degrees. 


90.56 


9.42 


35,100 


17.2 


458.0 


81.90 


17.99 


41,575 


27-5 


345 -o 


71.20 


28.54 


41,000 


24.0 


269.0 


60.94 


38.65 


48,520 


29.4 


202.0 


55-15 


44-44 


52,320 


32.7 


log.o 


49 66 


50.14 


43,154 


36.0 


38.0 


41.30 


58.12 


4,588 


100. 


1.8 


32.94 


66.23 


7,241 


100. 


1.2 


20.81 


77.63 


16,374 


100. 


0.8 


10.30 


88.88 


22,500 


85. 6 


7-1 


0.00 


100.00 


9,186 


38.1 


141. 5 



Timber. 



In the July, 1873, number of Van Nostrand's Magazine, 
Prof. Thurston gave the results of some experiments on timber 
test specimens of circular section, 0.875 inch in diameter. Eq. 
(12) may be written as follows : 



5-1 



(21) 



Prof. Thurston determined the values of C, and the values 
of Tni = 5. 1 (^ have been computed from them : 

T^ (per sq. in.) 

White pine 1,530 pounds. 

Yellow pine, sap 2,142 " 

Yellow pine, heart , 2,448 " 

Spruce 1,836 " 

Ash 2,632 '• 

Black walnut 3,366 " 



5IO TORSION. [Art. 60. 

Tm (per sq. in.). 

Red cedar 1.958 pounds. 

Spanish mahogany 35978 

Oak 3,244 

Hickory 5.202 

Locust 4, 8g6 

Chestnut 2, 142 

It is presumed that the axis of torsion was parallel to the 
fibres, which would cause the shear to take place across the 
latter. 

It is interesting to observe that 7^ is generally considerably- 
less than the ultimate resistance to simple shear as given in 
Table II. of Art. 59. 

If d is in inches and M in inch-pounds^ there may again be 
written : 



^ = 




\{ M is given in foot-pounds, 12M is to be written (or M. 
If Mj^ is the greatest working moment, and 7t the safety factor, 
nAfj is to be written for M, 



Relation between Ultimate Resistances to Tension and Torsion. 

In the '' Trans. Am. Soc. of Civ. Engrs.," Vol. VII., 1878, 
Prof. Thurston gave the results of some of his experiments 
which were made with a view to the determination of this re- 
lation. If J/ is the ultimate torsional moment \\\ foot-pounds 
of specimens one inch long and 0.625 inch in diameter ; d the 
angle of torsion corresponding to this greatest moment M ', 
and T the ultimate tensile resistance in pounds per square 
inch ; he deduced from a large number of steel specimens of 
wide range in grades the following formula : 



Art. 60.] TENSION AND TORSION. 5 1 1 



r^j/(9^^~^). 



300° 



3 

No experiments were made in which was greater than 
T is thus seen to increase as M increases and as d decreases. 



CHAPTER IX. 

Bending or Flexure. 

Art. 6i. — Coefficient of Elasticity. 

The coefificient of elasticity, as determined by experiments 
in flexure, can scarcely be considered other than a conven- 
tional quantity. If the coefficients of elasticity for pure ten- 
sion and compression were exactly equal to each other, and if 
all the hypotheses involved in the common theory of flexure 
were true, then, indeed, the coefficient of elasticity for flexure 
would possess actual existence, and be the same as that for 
either tension or compression. 

These conditions, however, never exist, and the quantities 
found in this chapter under the name " coefficient of elasticity" 
possess value chiefly as empirical factors which enable the 
deflections in the different cases to be estimated with sufficient 
accuracy for all ordinary purposes. 

The formulae to be used in the determination of the co- 
efficients of elasticity for flexure have already been established, 
and their use will be shown in succeeding Articles. 

Art. 62. — Formulae for Rupture. 

As with the formulae for the coefficient of elasticity, so with 
the formulae for rupture in bending ; they are all deductions 
from the common theory of flexure, and, strictly speaking, are 
subject to all the limitations involved in it. 

If K and K' are the greatest intensities of stress in the sec- 



Art. 62.] FORMULM FOR RUPTURE. 513 

tion of rupture and at the instant of rupture ; y the variable 
normal distance of any fibre from the neutral surface ; )\ and 
y the greatest values o{ y \ b the variable width of the section 
(normal toj); and J/ the resisting moment at the instant of 
rupture ; then the general formula for rupture by bending, as 
given by Eq. (i) of Art. 27, is : 



K 



M = 

y 



t fO 



■^1 X' 

y^b dy + , y'^b dy , , . , (i) 



y 



-y 



This equation is based on the supposition that the coeffi- 
cients of elasticity for tension and compression are not equal. 
Although this supposition is strictly true, yet equality is al- 
most invariably assumed ; particularly in the treatment of 
solid beams. Fortunately, this assumption is not far wrong in 
those materials which are most valuable to the engineer. 

Eq. ( I ), however, will hereafter be applied to some cast-iron 
flanged beams. 

If the tensile and compressive coefficients of elasticity are 

equal, • — = -— . Or, if K is the greatest intensity of stress in 

J'x y 
the section which exists in the fibre at the greatest normal 

distance, d,, from the neutral surface, tfien — = ---,and Eq.(i) 

yx ^i 

becomes : 

^ = f (^) 

This is Eq. (14) of Art. 18, and is the one almost invariably 
used in engineering practice. 

In Eq. (2) /is the moment of inertia of the cross section of 

the beam about its neutral axis. By introducing the value of 

/ for each particular shape of section, simple working forms of 

Eq. (2) may easily be obtained. This will be done for two 

sections in the following Article. 
33 



514 



FLEXURE OF SOLID BEAMS. 



[Art. 63. 



Art. 63. — Solid Rectangular and Circular Beams. 



While the rectangular form of cross section almost invari- 
ably characterizes timber beams, similar ones of iron, steel and 
other metals are only occasionally seen. Beams of iron and 
steel with circular cross sections, however, are quite common 
as pins in pin connection bridges. 

If ^Px represents the moment of the external forces about 
the neutral axis of any section, Eq. (2) of the preceding Article 
becomes : 



«1 



(I) 



The following are the values of / and d^ for rectangular and 
circular sections, h being the side of the rectangle normal, and 
b that parallel to the neutral axis, while r is the radius of the 
circular section, and A the area in each case : 





12 


12 


Rectangular: - 








T _ ^^"^ 


_ ^^ 


Circular : 


4 
d^^r. 


4 



If the beams are supported at each end and loaded by a 
weight W at the centre of the span (or distance between sup- 
ports), which may be represented by /, then the moment at the 
centre of the beam becomes : 



Art. 63.] WROUGHT IRON. 515 

Wl 

:epx = m= -— (2) 

4 

There will then result from Eq. (i) : 
For rectangular sections : 

,^ Wl KbJe KAh , , 

M — = — ^- = — >— (3) 

A 6 6 ^^^ 



For circular sections : 

,^ Wl TtKr^ KAr 
M = = = (4) 

4 4 4 ^^ 

The quantity iT is called the modulus of rupture for bending^ 
and if experiments have been made, so that W is known, Eq. 
(3) gives : 

K=^-Wl=^m (5) 



2 Ah 2 bh'' 



And Eq. (4) : 



K^^L = m^ (6) 

Ar Ttr^ ^ ^ 

If the rectangular section is square, bh^ = ^3 — /^s. 

Wrought Iron. 

If the beam is simply supported at each end and carries a 
load W 2X the centre, while E is the coefficient of elasticity and 
w the deflection at the centre, Eq. (8) of Art. 24 gives : 

T = i8£7 (7) 



5l6 FLEXURE OF SOLID BEAMS. [Art. 63. 

If, in any given experiment, w is measured, E may then be 
found by the following form of Eq. (7) : 

E^^L, (8) 

If the section is rectangular: 

77 ^^' ( \ 



Mr, Edwin Clark tested a one-inch square wrought-iron bar 
with the following results at the " elastic limit : " 

1—12 inches. W= 2,636.00 lbs. 

w = 0.09 inch. 3 = /i = I inch. 

Eq. (9) then gives : 

E = 12,652,809.00 pounds per square inch. 

The mean for 2 one and a half inches square bars was as 
follows : 

/ = 36 inches. W = 2,766.00 lbs. 

w = 0.305 inch. I? z=z /i =2 1.5 inches. 

.•. E = 20,894,600.00 pounds per square inch. 

A mean of 4 two inches square bars of Swedish iron, tested 
by Mr. Kirkaldy, in 1866, gave the following results at the 
'^ elastic limit : " 

/ = 25 inches. W = 6,625.00 lbs. 

w = 0.082 inch. b = A = 2 inches. 

E = 19,725,000.00 pounds per square inch. 



Art. 63.] WROUGHT IRON. 51/ 

By '* weighting " these results in proportion to the number 
of tests of which each is a mean, the mean of all becomes : 

E = 19,049,000.00 pounds per square inch. 

It is very probable that if w had been measured for smaller 
loads, E would have been materially increased. 

Mr. Kirkaldy tested the same four square Swedish iron bars 
to rupture. By the aid of Eq. (5), and the data given above, 
the greatest, mean, and least results were as follows : 

W. K. FINAL DEFLECTION. 

Greatest 15,885 lbs 74,475 lbs. per sq. in 5.85 ins. 

Mean 14,516 lbs 68,044 lbs. per sq. in 5.35 ins. 

Least 13,338 lbs 62,522 lbs. per sq. in 4.98 ins. 

The ultimate tensile resistance of the same iron was about 
45,000 pounds per square inch. These experiments would seem 
to show that K, for square bars under similar circumstances 
of span and depth, may be taken about 1.5 times the ultimate 
resistance to tension. 

The results in the following table were computed by the aid 
of Eq. (6), for some circular beams of " Burden's Best " iron, 
which were tested at the Rensselaer Polytechnic Institute in 
November, 1882. As beams cannot be actually broken under 
such circumstances, the '^ ultimate" value of iTwas taken with 
a final deflection of one to one and quarter the diameter. 

The '' elastic limit " is taken at that point beyond which the 
metal *' flows," and is indicated by the incapability of the spec- 
imen to hold up the scale beam beyond it, under a small in- 
crease of stress ; in other words, it is that point at which the 
specimen '' breaks down." 

These experiments show conclusively that *' ultimate" K 
decreases as the ratio of span over diameter increases, but they 



518 



FLEXURE OF SOLID BEAMS. 



[Art. 63. 



Circular Beams of " Burden's Best " Wrought Iron. 





DIAMETER. 


SPAN. 


w. 


K. 


KIND. 


Elastic. 


Ultimate. 


Elastic. 


Ultimate. 


Turned . . . 

Turned . . . 
Turned . . . 
Turned . . . 
Rough. . . . 
Rough. . . . 
Turned . . . 
Turned . . . 
Rough .... 
Rough. . . . 
Turned . . . 
Turned . . . 
Turned . . . 
Turned . . . 


Ins. 
1.25 
1.25 
1.25 

1.25 

1. 00 
1. 00 
1. 00 
1. 00 
1. 00 
1. 00 
0.75 
0.75 
0.75 
0.75 


Ins. 
12 

8 
12 

8 
12 

8 
12 

8 
12 

8 
12 

8 
12 

8 


Lbs. 
3,000 
4,400 


Lbs. 

6,000 

10,500 


Lbs. 
46,950 
45,900 
54,760 
52,150 
55,000 
57,000 
55,000 

51,950 
57,000 
47,100 
53,880 
47,100 
58,370 


Lbs. 

93,900 

109, 500 
93,870 

114,700 
91,700 

101,900 
91,600 

107,000 
91,680 
97,800 
74,050 
85,310 
74,050 
85,310 






















1,700 
2,800 

700 
1,200 

700 
1,300 


3,000 
4,800 
1,100 
1,900 
1,100 
1,900 



are not sufficiently extended to establish the limits of applica- 
tion of the observation. 

Cast Iron, 



All the following results for cast-iron beams are found from 
Major Wade's experiments (" Strength and other Properties of 
Metals for Cannon," 1856). His test bars were either two 
inches square in section or two inches in diameter, and were 
twenty-four inches long. They were loaded at the centre, and 
the distance between supports was twenty inches. The follow- 
ing table gives results for square bars. K is given in pounds 
per square inch, and is found by the aid of Eqs. (5) and (6). 
" Def.'' is the final deflection. 

Although Major Wade made many other experiments of 
the same kind, these may be considered representative ones. 



Art. 63.] 



CAST IRON. 



519 



Bars with square section, Eg. (5). 





HOURS 








KIND OF IRON. 


IN 

FUSION. 


W. 


K. 


DEF. 






Lbs. 


Lbs. 


In. 


r 





11,587 


42,130 


0.156 


Richmond iron ; 2d fusion J 


I 


12,487 


45,110 


0.152 




2 


15,019 


52,870 


0. 152 




2 


15,525 


55,930 


0.147 


Stockbridge iron ; 2d fusion j 


I 


11,812 


42,760 


0.162 


I 


14,512 


52,670 


0.195 






li 


16,481 


60, 500 


0.202 






2 


19,462 


69,680 


0.230 









12,987 


49,070 


0.250 









13,365 


50,120 


0.217 






I 


15,363 


57,330 


0.220 


Franklin iron ; 3d fusion -< 




I 
2 


14,616 

13,788 


54,550 
48,730 


0.195 




0.152 






2 


14,850 


50,720 


0. 170 






3 


16,056 


56,050 


0-T75 






3 


16,722 


60,410 


0.170 



Ba7-s with circular section, Eq. (6). 



Franklin iron ; 3d fusion ■< 



Franklin iron ; 2d fusion. 



3 

3l 

I 

2 

3 

4 



10,437 
8,665 
11,112 
10,606 
7,920 
9,270 
9,481 
7,920 



70,600 
57,720 
70,740 
71,740 
52,360 
63,670 
64,820 
52,360 



0.237 
0.166 
0.254 
0.240 



0.240 
0.262 



It is both interesting and important to observe that K and 
the final deflection are materially larger for circular beams than 
for square ones. 

By comparing these values of K with the ultimate tensile 
resistances found by Major Wade, and which have been given 
under the head of '' Tension," it will be seen that no great 
error will be involved if K is taken at twice the ultimate tensile 



D-^ 



FLEXURE OF SOLID BEAMS. 



[Art. 63. 



resistaiice for square bars, and two and a quarter times the same 
quantity for bars with circular section. 

Whether these ratios will hold for iron of inferior quality to 
that used by Major Wade, can only be determined by farther 
experimenting. 



Steel, 



Some circular Bessemer steel beams with 12 and 8-inch 
spans were tested at the Rensselaer Polytechnic Institute in 
Nov., 1882, with the results which are given in the next table. 
The "■ elastic limit " is that point at which the specimen 
*' breaks down." The " ultimate " value was that for which 
the deflection was equal to one or one and a quarter the 
diameter. 

Circular Bessemer steel beams, Eq. (6). 





DIAM. 


SPAN. 


w. 


K. 




Elastic. 


Ultimate. 


Elastic-. 


Ultimate. 




In. 
I.OO 
I.OO 
I.OO 
I.OO 
0.75 
0.75 
0.75 
0.75 


Ins. 

12 

8 

12 

8 
12 

8 
12 

8 


Lbs. 


Lbs. 


Lbs. 
86,000 
85,300 
76,400 
76,400 
77,400 
80,800 
77,400 
80,800 


Lbs. 
146,750 
152,800 
137,520 
152,800 
122,200 










2,500 
3,750 
1,150 
1,800 
1,150 
1,800 


4,500 
7,500 
1,800 
3.300 
1,700 
3,300 








148,200 
114,400 
148,200 









The " ultimate " K is seen to decrease as the ratio of length 
over diameter increases. 

The following table contains results computed from the ex- 
periments of the " Steel Committee " of the British Institu- 
tion of Civil Engineers ; the experiments were made in 1868. 



Art. 63.] 



STEEL. 



521 



The bars were 1.9 inches square in section, and the distance 
between supports was twenty inches. 

Bessemer Steel, Eq. (5). 



KIND AND NUMBER OF TESTS. 


K. 


ELASTIC OVER 
ULTIMATE. 


FINAL DEFLEC- 
TION IN INCHES. 


5, tires, hammered 


Lbs. per sq. in. 
129,030 
129,325 
125,900 
115,120 


0.573 
0.615 
0.612 
0.563 


3-82 

4.08 

3-94 

4-03 


5, axles, " 


4. rails. " 


4, tires, axles, rails ; rolled 



5, tires, hammered. 
4, axles, " 
I, rail, 
I, axle, rolled. . . . . 



Crucible Steel, Eq. (5). 



143^530 
152,055 
175,470 
118,160 



0.574 

•0-539 
0436 

0.538 



3.32 
3.35 
3-65 
3-84 



Each result is an average of the number of tests shown in 
the left column. 

The ratio " elastic over ultimate " is the value of K at the 
" elastic limit " divided by its ultimate value as given in the 
table. 

Table IX. of Art. 34 gives the ultimate tensile resistances 
of these same steels. That table, taken in connection with the 
results just given, shows that K is about 1.66 times the ultimate 
tensile resistance for square Bessemer steel bars ^ and about 1.85 
times the same quantity for square crucible steel bars, 

Mr. J. W. Cloud, of the Penn. R. R. Co., made bending 
tests of the Bessemer rail steel whose ultimate tensile resist- 
ances are given in Table IV. of Art. 34. His test pieces were 
12 inches long, 1.5 inches wide, and 0.5 inch thick. The load 
was applied in the direction of the thickness, and midway be- 
tween supports 10 inches apart. The greatest, mean and least 



522 



FLEXURE OF SOLID BEAMS. 



[Art. 63. 



results of the 18 means of the groups shown in Table IV., Art. 
34, are the following: 

w. K. 

Greatest 3,349 lbs 133,960 lbs. per sq. in . 

Mean 3,026 lbs 121,040 lbs. per sq. in. 

Least 2,765 lbs 110,600 lbs. per sq. in. 

With these rectangular specimens of Bessemer rail steel, 
supported flatwise, therefore, K may be taken about 1.6 the 
ultimate tensile resistance. 

The following table contains the results of Mr. Kirkaldy's 
experiments on square bars of Fagersta steel. These bars 
were 1.9 inches square in section, and rested on supports 20 
inches apart. W is the breaking weight at centre, and K is 



Fagersta Steel Square Bars. 





w. 




K, 




ELASTIC OVER ULTI- 


FINAL DE- 


MARK. 


















POUNDS. 


LBS. PER SQ. IN. 


MATE. 


FLECTION. 


I 2 


30,496) 




133,380) 




0.669) 


11 


Ins. 

75 


1.2 


32,896 [• 


« "2 


143,880^ 


0.695V 


gS. 


0.72 


1.2 


35,376) 


g CO 


154,710) 


s? 


0.616) 


UJ 


0.87 


0.9 


43,820) 


II CO 


191,640) 


"8 


0.500) 


11 . 


1.46 


0.9 


44,552 ^ 


§co 


194,850 > 


t m" 


0.476V 


c 


1.62 


0.9 


43,12s) 


s^ 


188,640) 




0.512) 




1.38 


0.6 


40,260) 
36,200 V 
38,120) 




176,100) 


lid 


0.467) 


II 


3-15 


0.6 


rt ". 


158,310 


?> r^ 


0.491 V 




3-56 


0.6 




166,740 




0.4S2) 


0) 


3.22 


03 


24,420) 


lU 


106,800) 


II d 

CO 


0.561) 


II . 


5.22 


0.3 


23,280 v 


rt <^. 


101,820 >■ 


Si 0" 


0.653 V 


si ^ 


5 -05 


0.3 


28,150) 


^ in 


123,120) 


§M 


0.654) 




5-05 



Art. 63.] COMBINED STEEL AND IRON. 523 

computed by the aid of Eq. (5). The column '''■Elastic over 
ultimate'' contains the ratios of the values of K at the '■'■ elastic 
limit " divided by the ultimate values given in the table. 

The '■^ Mark'' shows the character of the steel; 1.2 is the 
hardest, and 0.3 the softest. 

K is about 1.6 times the ultimate tensile resistance for the 
grades 1.2 and 0.6, and 1.8 times the same quantity for the 
grades 0.9 and 0.3. 

Combined Steel and Iron. 

In Sept., 1881, some interesting and valuable experiments 
on the transverse resistance of pins (solid circular beams) were 
made at Phcenixville, Pa., by the Phoenix Iron Co. 

The pins were of combined iron and steel, the core of the 
pin being of steel, and the outside of iron. In such a pin the 
iron seems to change gradually to the steel, but the shell of 
iron may perhaps be considered one quarter to one half an inch 
thick. 

These pins are supported at each end and loaJed in the 
centre. The results of the experiments are given in the fol- 
lowing table : 

D = diameter of pin. 

/ =: length in inches between supports. 

W — weight (pounds) at centre. 

K' = intensity of stress per sq. in. on extreme fibre, in 

general. 
K = intensity of stress per sq. in. on extreme fibre, at 

rupture. 

K is the greatest value of K' for any one pin. Either K or 
K'f by Eq. (6), has the value : 

Wl 

Ar 



524 



FLEXURE OF SOLID BEAMS. 



[Art. 63. 



PIN. 


Z>. 


I. 




Ins. 




I 


4.V 


24 


I 


4.V 


24 


I 


4*^6- 


24 


2 


4^ 


24 


2 


4i 


24 


2 


4^- 


24 


3 


4i 


20.5 


3 


4^ 


20.5 


3 


4i 


20.5 



w. 



36,000 
60,000 

100,000 
60,000 
84,000 

148,000 
68,000 
84,000 

252,000 



K' OR K. 



32,815. 
54,692. 

91,^54 
40,241. 

56,337. 
99,260. 

38,955- 

48,121. 

144,360. 



= K 



= K 



Elastic limit. 
Not broken. 

Elastic limit. 
Broken. 



Deflection 



ms. 



Slight permanent set. 
Broken. 



The mean of the two values of iT is : 



„ 99,260 + 144,360 „ J 

K = ^^ ' — -^-^^ — = 121,810.00 pounds. 



Copper^ Tin, Zmc^ and their A Hoys. 

I 
In the following table are given the data and the results of 

the experiments of Prof. R. H. Thurston, as contained in his 

various papers, to which reference has already been made. 

The distance between the points of support was twenty-two 

inches, while the bars were about one inch square in section, 

and of cast metal. 

The modulus of rupture, K^ is found by Eq. (5), in which, 
however, in many of these cases, W is the weight applied at 
the centre, added to half the weight of the bar. When K is 
large and the specimens small, this correction for the weight of 
the bar is unnecessary ; otherwise, it is advisable to introduce it. 

The coefficient of elasticity, E, is found by Eq. (9), in which 
W\^ the centre load added to five-eighths of the weight of the 
bar. 

The manner in which both these corrections arise, is com- 
pletely shown in Case 2 of Art. 24. 



Art. 63.] 



COPPER, TIN, ZINC AND AIIOYS. 



525 



E, for any particular bar, has a varying value for different 
degrees of stress and strain. Those given in the table may 
be considered average values within the elastic limit. 

As usual, " elastic over ultimate " is the ratio of K at the 
elastic limit over its ultimate value. 

An examination of the ultimate tensile and compressive 
resistances of these same alloys, as given in preceding pages, 
shows that the ratio of K over either of those resistances is 
very variable. It is usually found between them, but occasion- 
ally it exceeds both. 









Square 


Bars, 






PERCENTAGE 


OF 
















LBS. PER SQ. IN. 


ELASTIC OVER 

ULTIMATE. 


FINAL DEFLEC- 
TION. 


E, 








LBS. PER SQ. IN. 


Cu. 


Sn. 


Zn. 




















Ins. 




100 


0.00 


0.00 


29,850 




8.00 


9,000,000 


f 


100 


0.00 


0.00 


25,920 


j 0.14 

(to 0.41 


1-38 
to 8.00 


- 10,830,600 


100 


0.00 


0.00 


21,251 


0.346 


2.31 


13,986,600 


100 


0.00 


0.00 


29,848 


0. 140 


Bent. 


10,203,200 


90 


10.00 


0.00 


49,400 


0.400 


Bent. 


14,012,135 


90 


10.00 


0.00 


56,375 


0.41 


3-36 






80 


20.00 


0.00 


56,715 


0.657 


0.492 


13,304,200 


70 


30.00 


0.00 


12,076 


I. 00 


0.062 


15,321,740 


61.7 


38.3 


0.00 


2,761 


I. 00 


0.032 


9,663,990 


48.0 


52.0 


0.00 


3,600 


I. 00 


0.019 


17,039,130 


39-2 


60.8 


0.00 


8,400 


I. 00 


0.060 


12,302,350 


28.7 


71-3 


0.00 


8,067 


0.583 


0. 121 


9,982,832 


9-7 


90-3 


0.00 


5,305 


0.25 


Bent. 


7,665,988 


0.00 


100 


0.00 


3,740 


0.273 


Bent. 


6.734,840 


0.00 


100 


0.00 


4,559 


0.267 


Bent. 


5,635,590 


80.00 


0.00 


20.00 


21,193 




3-27 


1 1, COO, 000 




62.50 


0.00 


37-50 


43,216 




3-^3 


14,000,000 




58.22 
55-00 


2.30 
0.50 


39-48 
44-50 


95,620 

72,308 





1.99 


11,000,000 








92.32 


0.00 


7.68 


21,784 


0.30 


Bent. 


13,842,720 


82.93 


0.00 


16 98 


23,197 


O.4T 


Bent. 


14,425,150 


71.20 


0.00 


28.54 


24,468 


0.51 


Bent. 


14,035,330 


63-44 


0.00 


36.36 


43,216 


0.53 


Bent. 


14,101,300 


58.49 


0.00 


41.10 


63,304 


0.48 


Bent. 


11,850,000 


54-86 


0.00 


44.78 


47,955 


0.39 


Bent. 


10,816,050 



526 



FLEXURE OF SOLID BEAMS. 



[Art. 63. 



Square Bars, — Continued, 



PERCENTAGE 


OF 
















LBS. PER SQ. IN. 


ELASTIC OVER 
ULTIMATE. 


FINAL DEFLEC- 
TION. 


E, 








LBS. PER SQ. IN. 


Cu. 


Sn. 


Zn. 




















Ins. 




43.36 


0.00 


56.22 


17,691 


I. 00 


0.0982 


12,918,210 


36.62 


0.00 


62.78 


4,893 


I. 00 


0.0245 


14,121,780 


29.20 


0.00 


70.17 


16,579 


I. 00 


0.0449 


14,748,170 


20.81 


0.00 


77-63 


22,972 


I. 00 


0.1254 


14,469,650 


10.30 


0.00 


88.88 


41,347 


0-73 


0-5456 


12,809,470 


0.00 


0.00 


100.00 


7,539 


0.57 


. I 244 


6,984,644 


70.22 


8.90 


20.68 


50,541 




0.4019 


14,400,000 




56.88 
45- 00 


21-35 
23-75 


21.39 
31-25 


2,752 
6,5^2 




0.0146 
0.0150 


14,800,000 
7,000,000* 






66.25 


23-75 


10.00 


8,344 




0.0162 


12,000,000* 




10.00 


50.00 


40.00 


21,525 




Bent. 


9,000,000 




58.22 


2.30 


39-48 


95,623 




2.000 


10,600,000 




60.00 

65.00 


10.00 
20.00 


30 00 
15.00 


24,700 
11,932 




0.1267 
0.0514 


14,500,000 
17,000,000 






70.00 


10.00 


20.00 


36,520 




0.1837 


15,000,000 




75.00 


5.00 


20.00 


55,355 




Bent. 


13,000,000 


80.00 


10.00 


10.00 


67,117 




Bent. 


13,500,000 




55- 00 


5.00 


44-50 


72.308 




Bent. 


11,000,000 




60.00 


2.50 


37.50 


69,508 




1.500 


13,000,000 




72.52 
77-50 


7-50 
12.50 


20.00 
10.00 


51,839 
61,705 




Bent. 
0.705 


12,000,000 
13,500,000 






85.00 


12.50 


2.5 


62,405 




Bent. 


12,500,000 





* These bars were about half the length of the others. 



Timber Beams, 



As timber beams are always rectangular in section, Eq. (3) 
only will be needed. Retaining the notation of that equa- 
tion, if the beam carries a single weight W 2X the centre of the 
span /: 



„. 2 KAk 
W= — - 

3 / 



(10) 



If the total load W is uniformly distributed over the span : 



Art. 63.] TIMBER. S27. 

4KA/1 

W' = — - — ~, — (11) 

3 / ^ ^ 

As K IS supposed to be expressed in pounds per square 
inch, all dimensions in Eqs. (10) and (11) must be expressed in 
inches. 

In the use of timber beams it is usually convenient to take 
the span / in feet, and the breadth (J?) and depth (/?) in inches. 
Placing 12/ for /, therefore, in Eqs. (10) and (11); 

„, KAh , ,,,, KAh . . 

in which formulae / must be taken in feet and A and h in 
inches. 

If B be put for -— , Eq. 12 becomes : 
^ 18 

W=B~; and, W = 2B ^ . . . (13) 

Hence, when W and W have been determined by experi- 
ment : 

For single load W at centre : 



%i> 



7? ^'^ f ^ 

^ = -a7. (14) 



Ah 
For total load W uniformly distributed : 

^-YM (^5) 

If the beam has a section one inch square and is one foot 

W 
long, B — VV = — — . B, therefore, may be considered the 

unit of transverse rupture ; it is sometimes called the coefficient 
for centre breaking loads. 



528 



FLEXURE OF SOLID BEAMS. 



[Art. 63. 



Table I. is a condensed statement of the result of experi- 
n>ents by the late R. G. Hatfield, a complete account of which 
may be found in his *' Transverse Strains," 1877. All the test 

TABLE I. 



MATERIAL. 



K= iZB. 



Lbs. 
Georgia Pine 850 



Locust . . . . , 
White Oak. . 
Spruce . . . . . 
White Pine. 
Hemlock . . , 
Whitewood . 
Chestnut . . . 



1,200 
650 
550 
500 
450 
600 
480 



Lbs. 
15,300 

21,600 

11,700 

9,900 

9,000 

8,ioo 

10,800 

8,640 



MATERIAL. 



Ash 

Maple 

Hickory , 

Cherry , 

Black Walnut 

Canadian Oak. . . 
New England Fir 



B. 



Lbs. 
900 

1,100 

1,050 
650 
750 
590 
370 



K = 18^. 

Lbs. 
16,200 

19,800 

18,900 

1 1 , 700 

13,500 

10,600 

6,6ro 



specimens were of American woods with cross dimensions va- 
rying from one to two inches and span of 1.6 feet. 

Table II. contains the results of experiments on specimens 
of American timber, given by Prof. R. H. Thurston in the 
"Journal of the Franklin Institute," Oct., 1879. The test 
specimens were 3 inches square and 4.5 feet between supports. 
The coefficient of elasticity is in pounds per square inch, and 
is found by Eq. (9). 

Later experiments by Prof. Thurston (" Jour. Frank. Inst.," 
Sept., 1880), on a great variety of yellow pine specimens, both 
in respect to dimensions and degree of seasoning, induced him 
to draw the following conclusions in regard to that timber : 

The elasticity of yellow pine timber as used in construction 
is very variable, the coefificient varying from one to three mill- 
ions, the average being about two millions for small sections, 
and a little above one and a half millions of large timber. 



Art. 63.] 



TIMBER. 



529 



TABLE II. 
specimens "}, 1718. x '^ins. x \.Sft. 



MATERIAL. 



White Pine. .. 
Yellow Pine. . 

Locust 

Black Walnut. 
White Ash . . . 
White Oak . . . 
Live Oak 



ELASTIC 

LIMIT. 



Lbs. 

4,320 

12,720 
8,400 

5.640 

6,360 
7,200 

9.040 



K. 


B. 


Lbs. 


Lbs. 


5,280 


293 


16,740 


930 


13,680 


760 


7,440 


413 


9,720 


540 


9,840 ' 


547 


11,280 


627 



DEFLECTION IN INCHES. 



Elastic. 



0.86 
0.84 
0.82 
0.50 
1.50 

0.90 

0.94 



Ultimate. 



1.28 
1.96 
2 70 
0.72 
2.50 
1.76 
1.38 



COEFFICIENT 

OF 
ELASTICITY. 



Lbs. 

883,636 

3,534.727 
2,046,315 
1,944,000 
1,080,000 
1,620,000 
1,851,428 



The highest values are as often given by green as by sea- 
soned timber 

The density of the wood does not determine the coeffi- 
cient ; . . . . 

A high coefficient usually accompanies high tenacity and 
great transverse strength, but it is not invariably the fact that 
maximum ultimate strength is accompanied by initial stiff- 
ness 

iT varies from 10,000 to 17,000 pounds per square inch (or 
B, from 556 to 944) with a mean value of about 13,000 (or 
about 722 for B). 

In ''Van Nostrand's Magazine" for Feb., 1880, Mr. F. E. 
Kidder, B.C.E., gives the following results of experiments with 
5 yellow pine specimens about 1.25 inches square in section and 
8 white pine specimens about 1.5 inches square; all on sup- 
ports 40 inches apart : 
34 



530 



FLEXURE OF SOLID BEAMS. 



[Art. 63. 



Yellow Pine. 



GREATEST. MEAN. 

Coefficient of elasticity. . . 1,926,160 lbs 1,821,630 lbs. . , 

K. 14,654 lbs 13,048 lbs. . 

B 813 lbs 725 lbs.. 

White Pine. 

Coefficient of elasticity .. . 1,461,728 lbs 1,388,497 lbs. . . 

K. 9,440 lbs 8,297 lbs. . 

B 524 lbs 461 lbs. . 



LEAST. 

1,707,282 lbs. 

12,280 lbs. 

682 lbs. 



1,251,252 lbs. 

7,578 lbs. 

421 lbs. 



Table III. contains values of B which have been computed 
from data determined by MM. Chevandier and Wertheim 
(** Memoire sur les Propri^tes Mecaniques du Bois," 1848). The 
timber was from the Vosges. The great variations in the 
length of span and dimensions of beam render these especially 
valuable. 

TABLE III. 

Vosges Titnber. 



BREADTH. 


DEPTH. 


SPAN. 


W AT CENTRE. 


B. 


Ins. 


Ins. 


Ft. 


Lbs. 


Lbs. 




II. 4 


12.8 


42.64 


14,120 


339 




10. 


11.^ 


36.08 


11,867 


356 


. 


8.8 


9.6 


29.52 


7,584 


287 


fe^ 


6.7 


7-7 


29.52 


4,580 


355 




3-65 


4-85 


29.52 


1,137 


415 




9-7 


2.16 


9.91 


2,017 


445 




9-5 

r 9.2 


I. ir 


9.91 


581 


500 




10.9 


18.04 


17,356 


293 




8.6 


9-3 


18.04 


15,816 


392 




7.6 


8.6 


IS.O4 


11,495 


376 




6.3 


■7-4 


18.04 


12,155 


643 





5-4 


6.3 


18.04 


4,895 


421 


3.26 


3-2 


9.84 


1,188 


354 




3-07 


3.16 


8.20 


1,617 


433 




11.50 


2.15 


18.04 


957 


343 




5 64 


1.66 


9.84 


825 


532 




I 9-5 


I. II 


9.84 


715 


614 



Art. 63.] 



TIMBER. 



531 



The weights of the beams were allowed for in the manner 
already shown in that section of this Art. which is headed "■ Cop- 
per ^ Tifiy ZiiiCy and their Alloys'' 



TABLE IV. 

Laslett 's Tests. 

Sections 2x2 inches with span of 6 feet. 



KIND OF TIMBER. 



W^ IN LBS. 



Oak, English 

Oak, Engflish 

Oak, English 

Oak, French 

Oak, French 

Oak, Tuscan 

Oak, Sardinian 

Oak, Dantzic 

Oak, Spanish 

Oak, American, white 

Oak, American, Baltimore.. 

Oak, African (or teak) 

Teak, Moulmein ... 

Teak, Moulmein 

Iron wood, Burmah 

Chow, Borneo 

Greenheart, Guiana. . . 

Sabicu, Cuba 

Mahogany, Spanish 

Mahogany, Honduras 

Mahogany, Mexican 

Eucalyptus, Australia : 

Tewart 

mahogany 

iron-bark 

blue-gum 

Ash, English 

Ash, Canadian 

Elm. English 

Rock elm, Canada 

Fir, Dantzic 

Fir. Riga 

Fir, spruce, Canada 

Larch, Russia 

Cedar, Cuba 

Red pme. Canada 

Yellow pme, Canada 

Yellow pine, Canada 

Yellow pine, Canada 

Pitch pine, American 

Pitch pine, American 

Pitch pine, American ... 

Kauri pine, New Zealand 



562 
407 

813 

877 

831 
758 
758 
474 
562 
804 
723 
1,108 

913 

843 

1^273 

975 
1.333 
I1293 

856 

8?2 

783 

1,029 
686 

1,407 
712 
862 
638 

393 
920 

877 
600 
670 
626 
560 

653 
627 
483 
304 
1,049 

930 
744 
719 



2f, I.V LBS. 



422 
305 

6io 

658 
623 
569 
569 
356 
422 
603 
542 
831 
685 
632 
955 
731 
1,000 
970 
642 
602 
587 

772 
515 
1,055 
534 
647 

479 
295 
6go 
658 
450 

503 
470 
420 
490 
470 
362 
228 
7C7 
698 
558 
539 



FINAL DEFLEC- 
TION. 



Inches. 

5.10 

3-95 
7.71 
6.00 
7 58 
7.66 
6.50 
6.46 
6.62 
8.83 
7-13 
5-14 
3.38 
6.49 

4 25 
2.83 
4.62 
3-75 
3-45 
4.06 

3-92 

4-75 
4.71 
3-8i 
4.21 
8.63 

7-37 
5 29 
8.79 
5-14 
3-63 
5-19 
4-33 
4-37 
4.6} 
4.66 
3 39 
3-45 
4-79 
4.67 
4.42 
4.00 



COEFFICIENT 
OF ELAS., OR E. 



Pounds. 



902,600 

1,536,800 

1,440,000 

605,000 

871,400 



1,184,600 
1,547,200 
1,010,880 

1.378,500 
1,172.400 
2,369.300 
2,472,300 
1,057,900 
2,369,300 
1,882,800 
1,187,100 
2,021,800 

1,791.000 

2.420,000 
1,805,100 
1,404,000 



1,299,700 
1,395400 
1,763,200 
1,840,200 



2,030,800 
1,834,300 
1,602,000 
1,636,300 



E has been computed only for those cases in which W exceeds 700. 



532 FLEXURE OF SOLID BEAMS. [Art. 6^, 

In the cases of the fir specimens, B increases very con- 
siderably as the depth of the beam decreases, and with little 
irregularity. The same general result seems to hold with the 
oak specimens, although there are very marked irregularities. 
On the whole, therefore, these experiments would seem to 
show unmistakably that B ox K has much larger values for 
small depths of beam than large. 

The modulus of rupture, K^ may of course be found by tak- 
ing 1 8^, but its values are not given in the table. 

Tables IV., V., VI. and VII. contain values of B and E 
which have been computed from data determined by the 
English experiments of Messrs. Laslett, Maclure, Fincham, 
Edwin Clark and G. Graham Smith. These experiments are 
among the latest and most valuable ever made. 

In all these tables ^is the total load applied, including the 
weight of the beam, wherever that correction is made. 

In Table IV. the coefficient of elasticity is computed, in all 
cases, for a centre load of 390 pounds. In Table V. the centre 
load for the same computation is 1,680 pounds ; and in Table 
VII. the elastic load had different values for different beams. 

In all cases, except the four noted in Table VII., the ap- 
plied loads were placed at the centre of the span. 

Although these experiments do not embrace a great variety 
of cross section for all kinds of timber, yet Tables IV., VI. and 
VII. give much larger values of B for small depths of pine and 
fir beams than for large' ones. This is a very important con- 
sideration in connection with the ultimate resistance of beams, 
and probably obtains for all kinds of timber. In fact, Table 
III., as has been observed, indicates the same results for 
Vosges fir and oak. 

These experim-ents also showed that the coefficient of elas- 
ticity, E, varied materially in the same specimen for different 
deflections, and that values among the greatest may be found 
with large deflections ; also that the "elastic limit" for flexure 
in timber beams is more conventional than real, since with 



Art. 63.] 



TIMBER. 



533 



TABLE V. 
Fincham s Tests. 

3x3 inches, section ; 4 feet span ; very dry timber. 



KIND OF TIMBER. 


W. 


B. 


COEFFICIENT OF ELAS- 
TICITY. 


Riga fir 


Pounds. 
4,530 

3,780 

2,756 

3,292 

2,520 

4,110 


Pounds. 
670 

559 
408 

487 

373 
608 


Pounds. 

2,293,760 
1,593,000 
1,550,000 
1,850,000 
925,000 
1,977,400 


Red pine 


Yellow pine 


Norway fir 


Scotch, pine 


Kauri pine 





TABLE VI. 

Machires Tests. 
Specimens of Memel Fir. — 1849. 













FINAL DEFLECTION, 


BREADTH. 


DEPTH, 


SPAN. 


w. 


B. 




• 










INCHES. 


Inches. 


Inches. 


Feet. 


Pounds. 


Pounds. 




I 


I 


li 


483 


644 


0.75 


I 


I 


li 


450 


600 


0.75 


2 


2 


2f 


1,910 


637 


I. 00 


2 


2 


2| 


1,311 


437 


I. 125 


3 


3 


9 


1,104 


368 


3-5 


3 


3 


9 


1,482 


494 


4-5 


6 


12 


12 


34,720 


4S2 


2.0 


9 


12 


12 


38,080 


353 


2-5 


12 


12 


12 


61,600 


428 


3-25 



534 



FLEXURE OF SOLID BEAMS, 



[Art. 63. 



TABLE VII. 
Tests by Edwin Clark and G. Graham S?nith. 



American red pine 

American red pine 

American red pine . . . 

Memel fir 

Memel fir 

Baltic fir 

Baltic fir 

Pitch pine 

Pitch pine 

Pitch pine 

Pitch pine 

Red pine 

Red pine 

Quebec yellow pine . . . 

Quebec yellow pine. . . 

euebec yellow pine . . . 
uebec yellow pine. . . 



tt 










II 










< 

w 


DEPTH. 


SPAN. 


W. 


B. 


Inches. 


Inches. 


Feet. 


Pounds. 


Lbs. 


12 


12 


15 


33.497 


291 


12 


12 


15 


29,908 


260 


6 


6 


10.5 1 


Distnb'd 


256 


13. 5 


135 


68,560 


293 






j 
10.5 1 


Distrib'd 




13-5 


13.5 


68,560 


293 


6 


12 


12.25 


19.145 


271 


6 


12 


12.25 


23,625 


335 


6 


12 


12.25 


23,030 


320 


6 


12 


12.25 


23,700 


336 


14 


15 


10.5 


134.400 


448 


14 


15 


10.5 


132,610 


442 


6 


12 


12.25 


16,800 


238 


6 


12 


12.25 

J 
10.5 \ 


19,040 
Distrib'd 


270 


14 


15 


68,600 


229 






IO-5 1 


Distrib'd 




14 


IS 


68,600 


229 


14 


15 


lO-S 


85,792 


286 


14 


15 


10.5 


76,160 


254 



FINAL DE- 
FLECTION. 



Inches. 
4.00 
3.10 
1.68 



1-93 
1. 31 
1-31 
1. 14 



1.94 



COEFFICIENT 
OF ELAS. 



Pounds. 

1,443,830 
1,155,100 
1,015,900 

2,150,500 ~ 

1,561,300 
1,573,400 
1,442,300 
3,125,000 
1,431-300 
1,935,400 
1,693,403 
1,247,000 
1,247,000 

1,329,750 

1^329,750 
1,270,000 






loads about half the breaking weight, not only the deflection 
but the " set " varied with the time.- 

The quantity ordinarily termed the load at the " elastic 
limit " may be taken from 0.5 to 0.6 the breaking weight. In 
Table VII. it varied from 0.50 to 0.78. 

The latest experiments on timber beams are those pf Col. 
Laidley and Prof. Lanza; both experimented during 1881. 
Col. Laidley's results are given in Table VII<^. 

As was to be expected, in accordance with conclusions 
already drawn, the sticks of Oregon pine with the smallest 
depths gave values of K and B considerably larger than the 
others. These results emphasize the fact that for large beams 
K or B must be taken from tests on beams equally large if 
accurate computations are to be made. With these consider- 



Art. 63.] 



TIMBER. 



535 



TABLE Vila. 
Seasoned Sticks, Loaded at Centre. 



3 

4 
5 

6 

7 

8 

9 
10 



KIND OF WOOD. 



Oregon pine 

Oregon pine 

Oregon pine 

Oregon maple 

California laurel 

Ava Mexicana 

Oregon ash 

Mexican white mahogany. 

Mexican cedar 

Mexican mahogany 



2 

< 


WIDTH. 


DEPTH. 


Ins. 


Ins. 


Ins. 


44 


3 


48 


348 


22 


I 


22 


123 


22 


I 


21 


1.20 


44 


3 


63 


3-63 


44 


3 


58 


3-58 


44 


3 


69 


369 


44 


3 


64 


3-64 


44 


3 


77 


3.77 


44 


3 


75 


3-75 


44 


3 


75 


3.75 



K = 18^. 

LBS, PER SQ. 
INCH. 



11,900 
13,210 
16,570 
10,560 
8,920 

9.930 
8,460 
9,610 

7.935 
15.830 



B. 



661 

734 
921 

587 
496 
552 
470 

534 
441 

879 



Cross grained. 



Worm eaten. 



Cross gfrained. 



TABLE VII^. 
Seaso7ied Spruce Bea?ns. 













K = iSB. 




NO. 


SPAN. 


WIDTH. 


DEPTH. 


MANNER OF LOADING. 


LBS. PER SQUARE 
INCH. 


B, 




Feet. 


Inches. 


Inches. 








I 


15.00 


2.00 


12.00 


At centre. 


5,526 


307 


2 


6.60 


2.00 


9.00 


" " 


5,389 


299 


3 


15.00 


2.00 


12,00 




5,237 


291 


4 


6.67 


2.75 


9.00 




4,082 


226 


5 


4.00 


3.00 


9.00 




3,285 


183 


6 


10.00 


3.00 


9.00 




4,508 


250 


7 


15 .00 


3.00 


9.00 




5,651 


314 


8 


20.00 


3-90 


12.00 




4,253 


237 


9 


10.00 


2.50 


13 50 




3,787 


210 


10 


16.00 


3-75 


12.00 


4 . 5 feet from one end. 


3,271 


182 


II 


7.00 


7.00 


2.00 


At centre. 


8,748 


486 


12 


7.00 


1-75 


6 75 


" " 


7.562 


420 


13 


6.67 


3.00 


9 00 




4.931 


274 


M 


6.67 


3.00 


9 00 


At 4 points, 16 ins. apart. 


4,961 


276 


IS 


16.00 


3-90 


12.00 


4,5 feet from one end. 


S,2i8 


289 



53^ 



FLEXURE OF SOLID BEAMS. 



[Art. 63. 



ations in view, Prof. Lanza's results for large spruce beams, 
which are given in Table VI I<^., possess great value. 

With the exception of Nos. 11 and 12 the material was 
common merchantable lumber. 



Timber Beams of Natural and Prepared Wood. 

Table Vllr. contains the results of some experiments by 
A. M. Wellington, C.E. ('' R. R. Gazette," Dec. 17, 1880) on 
small specimens i^ inches square and 15 inches between sup- 
ports. ''AH the woods, except as specified, had been cut six 
to eight months and were partially seasoned." 

TABLE Vllr. 
Specimens 1.25 itiches square, 15 inches long. 



KIND OF TIMBER. 



White oak, well seasoned. . 

White ash , 

Beech 

Elm 

Pin oak 

White oak, green 

Soft maple 

Black ash 

Sycamore 



NATURAL. 


PREPARED. 


W, in Lbs. 


B^ in Lbs. 


W, in Lbs. 


B, in Lbs. 


989 


633 






926 


593 


825 


527 


864 


553 


801 


513 


763 


489 


763 


489 


941 


602 


755 


482 


747 


479 






742 


476 


643 


411 


685 


439 


640 


409 


628 


401 


550 


332 



LOSS, PER 
CENT. 



11.2 

7.2 

0.0 
20.0 

13-7 
6.9 

17.2 



The '' prepared " specimens had been treated by the Thil- 
meny (sulphate of baryta) process ; and all specimens of the 
same kind of wood were cut from the same stick. 

The column '' Loss " is the per cent, of loss caused by the 
preservative process employed. 



Art. 63.] CEMENT, MORTAR AND CONCRETE. 537 



Cement, Mortar a7id Concrete, 

Table VIII. and Table IX. contain values of K computed 
from data given by Gen'l Gillmore in his " Limes, Hydraulic 
Cements and Mortars," 1872. All the prisms were 2 inches 
square in cross section and 8 inches long, and were broken by 
the weight W, which was applied at the centre of a 4-inch 
span. K is computed by Eq. (5), all dimensions being in 
inches. The composition is shown in the tables. The pure 
mortars of Table VIII. were kept 24 hours in a damp place, 
and then immersed in salt water until broken. Nos. i, 2, 3 
and 4 were 59 days old ; the others, 320. As a rule, those 
which set under pressure were considerably stronger than the 
others. 

In Table IX., all the prisms set under a pressure of 32 
pounds per square inch, and were kept in sea water, after the 
first 24 hours, until broken. 

Many reliable experiments, such as those which follow, 
show that when masonry is built in a strictly first-class manner, 
its transverse resistance is very considerable. 

Table X. is taken from a paper entitled ** Notes and Ex- 
periments on the Use and Testing of Portland Cement," by 
Wm. W. Maclay, C.E., in the ** Trans. Am. Soc. of Civ. Engrs.," 

1877. 

The concrete prisms were six inches square in cross section 

and two feet long, and rested on supports one foot apart. W 

was applied at the centre of the span. If W^ is the weight of 

the prism whose length is equal to the span, Eq. (5) becomes : 

ir=3(r^+^^.y (,6) 

2 bh'^ ^ ^ 

in which b, h and / are to be taken in inches. 



538 



FLEXURE OF SOLID BEAMS. 



[Art. 6i. 



< 


















(0 

Ph 



rt 


rt 


nj 


is 


^ 


^ 


(fl 


tfl 


t/5 








o 


o 


O 


> 


> 


> 



a a 



^ ^ ^ <A ^ lA 



u o <u <u 
rt ■" '^ ■" 



it! 



rt 


nS 


rt 


m 


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^ 


IS 


^ 


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•a 


T3 


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Art. 63.] CEMENT, MORTAR AND CONCRETE. 



539 



TABLE IX. 



Section of Prisms 2 i?tches square. Supports 4 inches apart. 



KIND OF CEMENT. 



English Portland (artificial) 

Cumberland , Md 

Newark and Rosendale 

Delafield and Baxter (Rosendale) 

" Hoffman," Rosendale 

" Lawrence," Rosendale 

Round Top, Md 

Utica, 111 

Sheperdstown, Va 

Akron, N. V 

Kingston and Rosendale 

Sandusky, Ohio 

James River, Va 

Lawrenceville Manf. Co. (Rosendale) 

Sandusky, Ohio 

Kensington, Conn 

Lawrence Cement Co., " Hoffman " Brand, 
Round Top, Md 



A', POUNDS PER SQUARE INCH. 



Pure 
cement. 



1,152 
716 
63 T 
6.7 

637 
583 

549 
560 

573 
540 
416 



602 
716 
656 



I vol. cement. 
I vol. sand. 



945 
690 
420 

519 
456 

450 
567 
464 
489 
417 
348 
468 
683 

532 
684 
630 



1 vol. cement. 

2 vols. sand. 



713 
419 

375 
399 



422 
338 
453 
375 

479 



580 



In Mr. Maclay's experiments, since the span was twelve 
inches and the ends overhung six inches, /v was computed by 
the formula : 



2 bh'' 



12 



(17) 



Table XI. contains the results of some French experiments 
cited by Gen. Gillmore in his *' Limes, Hydraulic Cements and 
Mortars." The concrete prisms were of Boulogne Portland 
cement, about 5.91 inches square in section, and broken by a 
weight (W) at the centre of a span of about 31.5 inches. K- 
was computed by Eq. (16). 

Table XII. gives the results of trials of concrete prisms by 
Gen. Totten, in June, July and August, 1837, ^^^^ prisms hav- 



540 



FLEXURE OF SOLID BEAMS. 



[Art. 63, 



TABLE X. 

Concrete Prisms 6" x 6" x 2'. Supports i foot apart. 



T. 


T. 


DISPOSITION OF PRISMS AFTER 
BEING MADE 


w. 


K. 


Fahr. 
18° 

18° 
18° 
18° 
18° 
18° 

24° 

24° 

32° 
32° 


Fahr. 
40° 
40° 
40° 

98^ 
98° 
40° 

97' 
40° 
98° 


Placed in North River. 

Exposed outside 

Kept indoors 

Placed in North River. 

Exposed outside 

Kept indoors 

Exposed outside 

< < <( 

(< (< 


d 

£ 

M 

in 


Pounds. 
525 
775 
1,125 
175 
325 
750 
1,800 
800 

1,475 
700 


Pounds. 

44 
60 

94 
15 
27 
63 
150 

67 
123 

58 



All prisms were of Portland cement conci-ete ; i vol. cement, 2 vols, sand, 5 
vols, small broken stone. 

T = temperature of air w^hen concrete was mixed. 
T' = temperature of concrete when mixed. 

ing been made in Dec, 1836. The cement was from Ulster 
Co., N. Y. The Hme (slightly hydraulic) was from Fort 
Adams, R. I., where the tests were made. W (the centre 
breaking weight) and K are in pounds. 

These experimental results on the flexure of solid beams in 
cement, cement mortar and concrete, in connection with those 
of Gen. Gillmore on the adhesion of bricks and cement or ce- 
ment mortar, show that masonry beams may have considerable 
transverse resistance ; and such resistance may be an important 
element of strength in some arches or similar masonry struct- 
ure. It should be borne in mind, however, that such a conclu- 
sion is implicitly based on the assumption of perfect manipu- 
lation of the cement and mortar and the most conscientious 
care in laying the masonry. These ends were attained in the 
test specimens, but it is probably safe to say that such is not 
the case even in what is termed first-class masonry. 



Art. 63.] CEMENT, MORTAR AND CONCRETE. 



541 



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VO 


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r^Ococo o^M cncno ■^eno O^ci Ceo cnr^ 
tn IT) Tt T^j- (N o^ --to vo lo-^t-'tcncncncncncn 

M M M MM 




cn M \n\D 0^ rtco CO en OO O O '^fCO N M O m 
0< vO rfoo UTO •^O XT) Tt CO ^0 i-i O^ Oco 

0_ en CO 00 "^Tfenenc^Mcnciop-iMM 

M M M 


< c 

H 


r^Ovo wcoco M rj-iriM r^oenM -^--i-movo 
CO uioi^r^OO M r--<N Tt^o "?1- M i-i rn r^ 

OcOCO-i-X^r^'^-l-cnC4C<M>-iMMMMMM 
M 


u5 

u 

H 
U 
K 
U 



u 


•paonpojd aiajo 
-uoD JO auinioA 


oen vnMOO>-icnO-i-i-<u->cnfnxoencn 
voo rfMO-t-MO-^MO'i-MO-t^O 




> 
c 

a 

B 
c 
U 


•S3iq 
-qad 53^ 






UBjaoj^ 


1 M — ;«—;■'— 1-* M elirr— 15<' M e<|r;H^ m sjk— |SJ m h;«i-!*< m c^i-^^-Im 


CO 

V, 


s 


•paonpoad j-bj 
-joxu JO amnpTY 


Tt CJ IT) vO 

M M <y> 

M M M M M 


J2 

> 
c 

c 
_o 

*J 

'35 


a 
B 

6 


•aaiB^ 


a en CO U-) Ti- cj 
'^ cn en en en 

d d d d d 6 


'pUBS B3S 




•juaraao 


M H*» H« Hi" H« 'Ao 



542 



FLEXURE OF SOLID BEAMS. 



[Art. 63. 



is. 



On 






S3*o 'aniiq 
OO'Z 'puBg 
oo-i '^uaoiaj 



oo'o 'auiii 
oo'z 'pu'Eg 

00 •! 'JU3UI33 



oS-i 'pucg 

OO'I 'jU3ai33 



oS-i 'puiGg 



Sz'o 'aaiiq 
oo-i 'purg 

OO'I 'JU3UI33 



OC'I 'pUT^S 

001 'JU3UI33 



Sz'o 'auiiq 
oS-o 'pucg 

OO-I '1U3UI33 



Tt C> C^ 


r^ a> 1^ 


10 c^ 


uJ" i-T 



f<^ M 
























^ a 
























vO C^ 00 CO 

















































en 



IT) CO M 



O O t^ 

10 CO CO 

O *-< vn 



O -to CI 

CO CO U-) T^ 



vr> 0^ r-x 


00 CO XT) c> 


•i- w -^ 


00 M U-) ci 


i-i m 


r^ w c) 



M M VO O "i- 1000 

e^ i~s.o CO in o t^ 
r-^ i-H CO CO M M M 



r^o O vo M o) a> 00 i^ f-^ f^ 
O coc^r^c^o-iM r^ 



O M 00 CO i^ C> N \r\^ o m 01 
CO loco w M u-)covo vnr^o^oo 



M r^co coo M 

rt O LO f^ M 1^ 

M rt l-( 1"^ M 



00 -t- -t r-> 
r^ w O CO 



O r^ O 00 tr> r^oo "^J- O co n r-^ o o 
M r^ Tf i^ vnvo 00 o^QO •^ M 0) vo CO 
coMC^ \Oi-iO»-(rf"coO»-icoM 

cf m" n CO XT) cT • cT 



vO 
X 



VO 



^ 



oo-o 'auiiq 
oS-o 'puijg 

oo-i 'lU3ai33 



OO'O 'ami';! 
OO'O 'puTjg 
001 '}U3ai33 



0> O CO o» r^ CO r->vO OO t^r^cor^Ncooo \r,\r, 
"^^O CO i-H M CO Tf -^ -^O rJ-vO oocOi-iir)'+i-iO-i- 
MMOcoi-'wOcoO c^O)i-icor^cocoiHcOM 



tOMOo ina "!fvr)0 r^C^i^r^i^M vni->.coo "^co 
r-«MO u^-fO O r^ O^^O Tt-^coTfci r^r-»o coO 
O^coOWWWoomO cOMT:fcoOcoo)M\OiH 



^:i<;^^^l<^l^^^^!^^!i<^^^^^^ 



rt . i3 . c« . i5 



<t_i c 
O b/3 

c c3 



C c cr y (/; <-! 



O mO 



OJ I-* 

£'- 



a . is 
o • o 

£ : s 



Tt OTj-T^-r. 



05 



<U rt 1^ 

1_ v3 1— 



« 



^ £ 
o ta 

c3 



?! 2f rt Ji rt 5^ u-^ 
!^<i; ^ bfl^ tyj^ b/)^ b/)^-" 



C3 1) rt O f^ 
-■ "" -- "" (U 



^ 



^ 



<u o 



£.^£|£|£'S£'H£SSo|S) 



MWe«c/2MC/2NpQMPQwc/2 



O! 



Art. 63.] STONE BEAMS. 543 

Sto7ie Beams. 

But few experiments have been made on the transverse re- 
sistance of the different kinds of stone. The following values 
of K have been computed from the experiments of R. G. Hat- 
field (" Transverse Strains ") and Gen. Gillmore ('* Building 
Stones"). 

B. K = 18B. 

Blue Stone Flagging 200 lbs 3,600 lbs. ') 

Sandstone 59 lbs 1,062 lbs. 

Brick, common 33 lbs 594 lbs. V Hatfield. 

Brick, pressed 37 lbs 666 lbs. 

Marble, Eastchester 147 lbs 2,646 lbs. ^ 

Granite, Millstone Point (doubtful). . . . 133 lbs 2,390 lbs. "j 

Marble, Eastchester 128 lbs 2,300 lbs. |- Gillmore. 

Granite, Keene, N. H 103 lbs 1,860 lbs. J 

All beams were broken by centre weights. The last three 
tests were with prisms 2 ins. X 2 ins. X 6 ins., over a span 
which was taken at 3 inches. 

Practical Formula; for Solid Beams, 

The quantities B, K and E^ which have been established, 
form a practical basis on which the deflection and ultimate 
resistance of solid beams are to be computed. 

Breaking weight {in potuids^ at centre of circular beam^ Eq. 
(6): 

^^KAr^nr^K ^^g^ 

^ If 

If W\^ 2, uniform load : 

T;rr 2KAr 27rr'^K , v 

^=^-=-7- ('9) 

In Eqs. (18) and (19), A (the area), r (the radius) and / (the 
span) are to be taken in inches. 



544 FLEXURE OF SOLID BEAMS. [Art. 63. 

Breaking weight (in pounds) at centre of rectangular beamSy 

Eg, (5) : 

^., 2 KAh 2 bh^K , . 

W= ^ = J- (20) 

If Wis 3. uniform load : 

„, A KAh AMfK f . 

^^-—7- = -^— (21) 

3^3^ 

In Eqs. (20) and (21), A (the area), b (the breadth), d (the 
depth) and / (the span), are to be taken in inches. 

If / is expressed in feet, and all other dimensions in inches, 
Eq. (20) becomes : 



and Eq. (21) : 



W= B^= B^J^ (22) 



TX7 * r. -Ah „ bh^ f V 

W = 2B -j~ = 2B -J- (23) 



(24) 



Deflection {in inches) at centre of circjclar beams : 

\2Enr^ \2EAr' 

Deflection {i?t inches^ at the centre of rectangular beams : 

'^- '^.Ebh^ - ^ZZ7. • • • ^^^ 

In Eqs. (24) and (25), Wis the centre load, and //the total 
uniform load, expressed in pounds; while A (area), /^ (cube of 
span), r (radius), b (breadth), and d (depth), are to be taken in 
inches. If there is no uniform load,// is zero ; and if there is 
no centre load, Wis zero. 



Art. 63.] COMPARISON OF MODULI. 545 



Comparison of Modulus of Rupture for Bending with Ultimate 

Resistances. 

The experiments on solid beams which have been cited, 
show the somewhat remarkable result that, in general, K has 
neither the value of the ultimate resistance to tension nor of 
that to compression ; nor, indeed, in some cases, is there any- 
thing like a well defined relation between those quantities. If 
those ultimate resistances have widely different values, K may 
be found between them ; in other cases it may considerably 
exceed either. In no case, however, it may safely be asserted, 
will it be found less than both. These investigations show that 
K varies with the kind of cross section, and it is altogether 
probable that it also varies with varying proportions of the same 
kind of cross section. Experimental data for the determina- 
tion of this point, however, are still lacking. 

In the absence of experiments conducted in a manner proper 
to the solution of this problem, it is difficult to assign confi- 
dently the reason for the facts as they appear. 

The explanation will probably be found in the effects of the 
following causes, while it is borne in mind that with the small 
ratios of span to depth usually found in connection with solid 
beams, the common theory of flexure is only loosely approxi- 
mate, and hence, that the greatest intensity shown by the 
common formulae is probably considerably different from the 
actual. 

The varying intensity in adjacent fibres prevents perfect 
freedom in lateral strains, and causes a corresponding increase 
in resistance. In the experiments which have been made, the 
place of greatest intensity of stress is exceedingly small, thus 
placing the part first ruptured somewhat in the condition of a 
very short specimen. Again, after the elastic limit is passed, 
in consequence of the flow of the material, it is highly proba- 
ble that the law of the variation of stress intensity changes and 
35 



546 



UNEQUAL-FLANGED BEAMS. 



[Art. 64. 



becomes such that, with the same greatest intensity at the sur- 
face of the soHd beam, the resisting moment is considerably 
increased. 

Finally, it has been shown that the experimentally deter- 
mined ultimate resistances to tension and compression are, 
in reality, mean intensities, and not the greatest which the 
material is capable of exerting at any one point, or along any 
one line, as in the extreme fibres of a bent beam. On this 
ground alone, K ought to be considerably greater than either 
T or Cy as determined from the usual cross sections. 



Art. 64. — Flanged Beams with Unequal Flanges. 



In the beams which are to follow, the material is distributed 
in a much more advantageous manner, in respect of its resist- 
ing moment, than in the solid beams which have been hereto- 
fore treated. In these beams, it will be found, in almost all 
cases, that the ultimate intensity of bending stress, at the point 
which first ruptures, is equal either to the ultimate resistance 
to tension or compression. In this respect, at least, therefore, 
the ultimate load for flanged beams is more easily and exactly 
determined than for solid ones. 

In Fig. I is shown a "flanged beam." The "flanges" are 

the two horizontal parts above and 
below; the "web" is the vertical part 
uniting the two flanges so as to form 
the perfect beam. 

In order that there may be economy 

A j_c^J ' B of material in the beam, neither flange 

)^ must begin to fail before the other ; in 
other words, the two exterior layers of 
fibres, above and below, must begin to 
fail at the same time. 

The intensities, then, in these two 



|^-__6 ^ 



> I) 



I 



r- 



Fig.l 



-fe ^ 



F 



Art. 64.] EQUAL COEFFICIENTS OF ELASTICITY. $47 

layers must, at the instant of rupture, equal the ultimate re- 
sistances to tension and compression in bending. 



Equal Coefficients of Elasticity, 

By the common theory of flexure, if the two coefficients of 
elasticity are equal, it has been shown that if C is the centre of 
gravity of the cross section, the neutral axis of the section will 
pass through that point. Let it now be supposed that the 
lower flange is in tension while the upper is in compression. 
Also let 7^ represent the ultimate resistance to tension in bend- 
ing, and let C represent the same quantity for compression in 
bending. Then, since intensities vary directly as distances 
from the neutral axis, 

J=f= •■• K = h'L = rih ... (I) 

The ratio between ultimate intensities is represented by 
v! . \{ d — h -\- h^ is the total depth of the beam, and hence 
h = d — hj'. 



dn' \^ 
K = n'(d - //,) = -^-^ = ^ .... (2) 



If, as an example, for mild steel there be taken : 

n' = ~ = 0.75 ; /t, = ±. 2 d= ^ . 
C '"' ' 7 4 7 

The relation between h and /'^ shown in Eq. (2) is entirely 
independent of the form of cross section. But according to 
the principles just given, in order that economy of material 




54^ UNEQUAL-FLANGED BEAMS. [Art. 64. 

shall obtain, the cross section should be so designed that h and hi 
shall represent the distances of the centre of gravity from the ex- 
terior fibres. 

The analytical expression for the distance of the centre of 
gravity from DF is: 

_ y2b'd^ -\-{b - b')t\d - v^t') ^y^{b^- b')t,^ , 
-^^ ~ b'd + (^ - by + {b, - by, • • ^^^ 

The meaning of the letters used is fully shown in the figure. 
In order that the beam shall be equally strong in the two 
flanges, the various dimensions of the beam must be so de- 
signed that 

x,^ K (4) 

It would probably be found far more convenient to cut sec- 
tions out of stiff manilla paper and balance them upon a knife 
edge. 

The moment of inertia about the axis AB, thus deter- 
mined, is : 

/ = %\blfi + bji} -{b- V){h - tj - {b, - b'){K - ty\. 

This value is to be substituted in Eq. (2) of Art. 62, now 
changed to 

CI TT 

For various beams whose lengths are / and total load W, 
the greatest value of M becomes : 

Cantilever uniformly loaded : 

Wl 
2 



Art. 64.] EQUAL COEFFICIENTS OF ELASTICITY. 549 

Cantilever loaded at end : 

M = Wl. 
Beam supported at each end and uniformly loaded : 



M 



Wl _ pi' 



8 8 



Beam supported at each end and loaded at centre 

Wl 
4 

The last two cases combined : 



^^i(K±Pl). 



Sometimes the resistance of the web Is omitted from con- 
sideration. In such a case the intensity of stress in each flange 
is assumed to be uniform and equal to either 7" or C. At the 
same time the lever arms of the different fibres are taken to be 
uniform, and equal to h for one flange and //j for the other, h 
and h^ now representing the vertical distances from the neutral 
axis to the centres of gravity of the flange s^ while d ^= h -\- h^. 

On these assumptions, if f is the area of the upper flange, 
and/' that of the lower, there will result: 

M = fC.h+fT.K (5) 

But since the case is one of pure flexure : 

fC = fT (6) 

.-. M = /C{h -\- h,) = fCd = f Td ... (7) 



550 



UNEQUAL-FLANGED BEAMS. 



[Art. 64. 



Also, from Eq. (6) : 



f c 



(8) 



Or, the areas of the flanges are inversely as the ultimate re- 
sistances. 



Unequal Coefficients of Elasticity, 

All these results presuppose equality between the coeffi- 
cients of elasticity for tension and compression. In some cases 
this presumption is not permissible. To the formulae of Art. 
27 resort must then be made. 

The neutral surface must first be located. If d is the total 
depth of the beam, h^ = d — h ', h, then, must be found. Eq. 
(5) of Art. 27, when applied to Fig. i; becomes : 



E' 



-bh' _ {b - b') [h - tj 

2 2 



= £ 



-bid - hy 



{K - b') {d-h- t,y 



E representing the coefficient of elasticity for compression, and 
E that quantity for tension. 

Performing the operations indicated and reducing, writing 
n for E' -^ E : 

{n - i)b'h' + 2int\b - b') + tlb, - b') + b'd'\h 

= nt\b - b') 4- {2d - /,) {b, - by, 4- b'd' , . (9) 

h is to be measured on the compression side of the beam. 
This is a quadratic equation of condition for the determina- 
tion of h. It is best to leave it as it is until the numerical sub- 



Art. 64.] UNEQUAL COEFFICIENTS. 551 

stitutions are made and then to solve it. //^ immediately results 
from the equation h^ = d — h. 

Frequently there is no compression flange, the section being 
like that shown in Fig. 2. In such a case b is 
equal to b\ or f is equal to zero ; hence the two 
terms nt\b — b') and nt'\b — b') in Eq. (9) disap- 
pear. No other change occurs. , 



Eq. (i) of Art. 27 then gives the following ^^ — ~ 

resisting moment of the section : 



C /,,. „ ,„ ., .„. , d,/i,3 






n 



n 



(10) 



C is the greatest intensity of stress in the section of the 
same kind as E . 

If the section is like Fig. 2, b again equals b' and the term 
{b — b') {h — t'Y in Eq. (10) disappears, but nothing else is 
changed. 

If 7" is the greatest stress on the other side of the neutral 
surface from Ci 



T 

M 



p-^ [nbh^ - n(b - b') {h - ty + bA' 

-{b,- b'){h,-t,y-] (II) 



In order that the beam may be equally strong in the two 
flanges, the ratio between h and h^, as determined by Eq. (9), 
should be the same as that determined by the following proc- 
ess. If ?/ is the rate of strain at units' distance from the neu- 
tral surface : 



552 UNEQUAL-FLANGED BEAMS. [Art. 64. 

Euh = C\ j^ Q^ 

\ ''' ~/7 ~ ^TF~' • • • • (12) 

If there is no waste of material, the cross section must be 
so designed that the ratios given by Eqs. (9) and (12) will be 
the same. 

If the thicknesses of the flanges /' and t^ are small com- 
pared with the depth d of the beam, and if b' also is small, i. r., 
if the flanges are assumed to give the wJiole resistance to bend- 
ing while the web takes up the shear, Eqs. (10) and (11) may 
be much simplified. 

. C T 
Making, therefore, ^' = o in Eq. (10), putting - — =. y- and 

then expanding the quantities (// — ty and [k^ — t^y : 
M = aC [a - / + g) + TiA {/h - /. + p 

Under the conditions taken, Cd^' = TbJ. ; also, — r and —'- 
are very small and may be neglected. Hence, 

M = Cbt' {d -t' - t,) = Tb,t, {d - t' - t,) . (13) 

But both of these approximations have made M too small. 
As an approximate compensation, therefore, — f — — — ^] may 
be written for — (/' + A)- The moment then becomes : 

M = Cbt' (d - ^L^^^-^ (14) 



The quantity within the parenthesis of the second member 
of this equation is evidently the distance between the centres 



Art. 64.] UNEQUAL COEFFICIENTS. 553 

of gravity of the flanges, while the quantity Cbt' = Tb^t^, is 
simply the flange stress. Eq. (14) is, then, identical with Eq. 
(7), as was to be anticipated. The equality of flange stresses 
gives : 

b,t, ~ C' 

a relation identical with Eq. (8). 

If desirable, an approximate correction for the neglect of 
the web may be introduced in Eq. (14). It has been seen that 
that equation is precisely the same as if B' were equal to ^, 
I.e., as if the two coefficients of elasticity were equal. Now, 
it will be shown in the next Article that U £' = E, the re- 
sistance of the web to bending is equal to that of one-sixth of 
its area of normal section concentrated in each flange. Hence, 
if a is the area of the normal section of the web, since b^' and b^fj^ 
are areas of the normal sections of the upper and lower flanges, 
there may be approximately written : 

,r(v, + |)(^-'-4i) . . . (.5) 

Values of C and 7" may be determined by experiment. 

In the case of solid beams, it has been seen that if r and r' 
are certain ratios, K = rT or r'C, Hence, since the web of a 
flanged beam is really a solid beam subjected to flexure, Eq. 
(15) may be written : 



M= TD {^' + ^) = CD (^^" + — ) . . . (16) 



In which. 



554 UNEQUAL-FLANGED BEAMS. [Art. 64. 

D = d — ^ = depth between flange centres ; 

2 

a' = ^j/j = area of bottom flange ; 

a" = bt' = area of top flange. 

Cast-Iro7i Flanged Beams. 

In the preceding Article it has been seen that r is equal to 
about 2 for a solid bar with square cross section. This would 
make r -f- 6 = ^. A few imperfect experimental indications, 
however, seem to indicate a decrease of r for a greater ratio of 
depth to breadth. Let, therefore, r -j- 6 = 0.25. Eq. (16) 
then becomes: 

.^= rz)(«+^) (17) 

li W = centre breaking load in pounds ; 

W^ = total uniform breaking load in pounds ; 

/ = span in feet ; 
12 / = span in inches : 



W ' 12/ 



s= 2lV/= TD (a' + -") : 



TD (a' + ^ 
/. W= —-. — ii- (18) 



3/ 
In the same manner 



m {a + ^) 



W^. = 2 — ~^r^ 09) 

Or, if// is the weight of the beam, supposed uniformly dis- 
tributed. 



Art. 64.] ■ CAST IRON. 555 

It has been shown under the head of *' Tension " that T 
varies from 15,000 pounds per square inch, for ordinary cast- 
ings, to 30,000 for those of extra quality. In Eqs. (18), (19) 
and (20), 

D must be taken in inches ; 

a and a in square inches ; and 

/ in feet. 

Those equations have been verified in a most satisfactory 
manner by the numerous English experiments of Hodgkinson 
and Cubitt (" Experimental Researches," etc., by Eaton Hodg- 
kinson, F.R.S., 1846), and Berkley (" Proc. Inst, of Civil 
Engineers," Vol. XXX.), as is shown by the following table. 
This table gives the actual centre breaking weights W, of the 
different beams, together with the values of W computed by the 
formula of Mr. D. K. Clark ('' Rules, Tables and Data"), which 
is essentially identical with Eq. (18); Mr. Clark taking the 
total depth minus the depth of the lower flange instead of 
"Z>," and " 0.28^," or " 0.29^," instead of " 0.25^." 

As the results are given to confirm the accuracy of the for- 
mulae under consideration, they are stated in tons of 2,240 
pounds. Nos. 17, 27 and 34 were of the form shown in Fig. 
2 ; the others had sections like Fig. i. The results for those 
three beams are not satisfactory, and Eq. (10) should therefore 
be used in all such cases where anything more than a very 
loose approximation is desired. In that Eq. n may be taken 
equal to unity, on account of the great irregularities in the 
ratio of the two coefficients of elasticity. Since, in this case 
(see Fig. i), b = b' Eq. (10) becomes : 

M=~ ibl0 + 3./..3 - {K - b') {K - m • • (21) 
3« 



556 



UNEQUAL-FLANGED BEAMS. 



[Art. 64. 



Cast-iron Flanged Beams. 









PROPORTION, UPPER 


COMPUTED 


ACTUAL 


NO. 


SPAN, 


CENTRE DEPTH. 














FLANGE TO LOWER, 


W (tons). 


^(TONS). 




Feet, 


Inches, 








I 


4-5 


5-125 


I to I 


2.47 


2.98 


2 


4 


5 


5 


125 


I to 2 


3-27 


3-29 


3 


4 


5 


5 


125 


I to 4 


3.83 


3-69 


4 


4 


5 


5 


125 


I to 4 


3.S7 


3-64 


5 


4 


5 


5 


125 


I to 4.5 


4.68 


4.79 


6 


4 


5 


5 


125 


I to 4 


6.45 


6.46 


7 


4 


5 


5 


125 


I to 5.5 


7.85 


7-47 


8 


4 


5 


5 


125 


I to 3.2 


6.49 


6,71 


9 


4 


5 


5 


125 


I to 4.3 


8.04 


7-54 


10 


4 


5 


5 


125 • 


I to 5.6 


9-56 


8.68 


II 


4 


5 


5 


125 


I to 6 


10. 98 


11.65 


12 


4 


5 


5 


125 


I to 7 


11.00 


10.40 


13 


4 


5 


5 


125 


I to 6.7 


9.02 


9.40 


14 


7 





6 


93 


I to 6 


10.26 


9.90 


15 


7 





4 


10 


I to 6 


5-41 


6,05 


16 


9 





10 


25 


« I to 8.3 


13.28 


12.80 


17 


4 


5 


5 


125 


none 


383 


3.93 


18 


4 


5 


5 


125 


I to 4 


9.67 


10.00 


19 


4 


5 


5 


125 


I to 4 


9.67 


10.00 


20 


4 


5 


5 


125 


I to 5-5 


11.85 


11.75 


21 


4 


5 


5 


125 


I to 5.5 


11.85 


11.85 


22 


4 


5 


5 


125 


I to 7 


16.47 


14.25 


23 


4 


5 


5 


125 


I to 7 


17.08 


18.00 


24 


18 





17 





I to 4.6 


24.93 


25.00 


25 


n 


67 


9 





I to 1,33 


21.24 


20.00 


26 


27 


4 


30 


5 


I to 2.1 


94.64 


76.60 


27 


23 


I 


36 


I 


none 


330.00 


153.00 


28 


15 





7 


15 


I to 3.6 


7.75 


7.00 


29 


15 





7 


17 


I to 3 . 6 


7.96 


7-13 


30 


15 





10 


75 


I to 2.3 


11.02 


11.50 


31 


15 





10 


75 


I to 2.3 


II. 71 


12,00 


32 


15 





12 


75 


I to 2 . 7 


11-95 


10.25 


33 


15 





12 


8 


I to 2.25 


14.89 


15.75 


34 


15 





14 





none 


18.39 


12.38 


35 


15 





17 


25 


I to 2 , 2 


19-39 


16.00 


36 


7 


5 


7 


15 


I to 3.4 


15^3 


15.63 


37 


7 


5 


10 


75 


I to 2,25 


21.76 


23.87 



If the weight of the beam is taken into consideration, as in 
Eq. (20) : 



Art. 64.] CAST IRON. 557 



M= {wj^fl\^l^ 



A mean of three of Mr. Hodgkinson's beams of 4.5 feet 
span, 5.125 inches depth, gave: 

H 
W -\-^— — S,y66 lbs., and C = 45,700 lbs. 



One of Mr. Cubltt's beams of 15 feet span and 14 inches 
depth, gave : 

W -\-^— — 28,100 lbs., and C — 30,850 lbs. 



The bottom flange of this beam was unsound : 
C must necessarily depend upon the span, since that portion 
of the web which is subjected to compression is somewhat in 
the condition of a long column. This, indeed, is true of the 
compression flange of any flanged beam, but the effects result- 
ing from such a condition are much more marked in the class 
of beams shown in Fig. 2. 

If, then, W is the centre breaking weight and W^ the total 
uniform breaking load (not including the weight of the beam), 
Eq. (21) becomes: 

^= •? = ^. ^''''" + ^■^''' - ^^- - ^'^ ^''^ ~ ''^'^ ~ 4 • ^^^^ 

In this equation, / must be taken in feet and other dimen- 
sions in inches. 

For 5 feet span 6' may be taken at 45,000 lbs. 
For 15 feet span ^may be taken at 35,000 lbs. 



55^ UNEQUAL-FLANGED BEAMS. [Art. 64. 

In order that a beam with top and bottom flanges may give 
the best result, z>., reach its ultimate resistance in each flange 
at the same time, Mr. Hodgkinson found that the area of the 
lower flange section should equal about six times that of the 
upper. That relation has been anticipated in Eq. (8). 



Deflection of Cast-Iron Flanged Beams, 

If W is the centre load in pounds, / and w the span and 
centre deflection, respectively, in inches, and / the moment of 
inertia of the cross section, Eq. (8) of Art. 24 gives : 

^ = 4W • • (^^) 

Or, if / is in feet, which is more convenient : 

A mean of two of Mr. Berkley's beams gave : 

/ = 4.5 feet ; w — 0.284 inch ; W = 20,160 lbs. : 
/= 18.74. Hence: E — 12,424,600 lbs. 

A mean of two of Mr. Cubitt's beams gave : 

/= 15 feet; w = 0.465 inch; W = 1 1,200 lbs. ; 
/= 227.03. Hence: £ = 12,886,720 lbs. 

The four preceding beams had top and bottom flanges, as 
in Fig. I. Another of Mr. Cubitt's beams, without top flange, 
as in Fig 2, gave : 



Art. 64.] WROUGHT IRON. 559 

/ = 15 feet ; w = 0.41 inch ; W = 13,440 lbs. ; 

/ = 2,22)' Hence : E = 10,679,400 lbs. 

This last beam had a defective bottom flange, hence there 
may be taken without essential error: 

£ = 12,000,000 lbs. 
Taking I in feet ^ Eq. (24) now gives for the centre deflection : 



w = -^ — ■ — ^ (25) 

1,000,000/ ^ ^^ 



in which W is either the centre load, or five-eighths (^^ths) the 
total uniform load, as the case may be. 

The formula by which /is to be computed is the one which 
immediately follows Eq. (4). 



Wrought-Iron T Beams. 

The wrought-iron X beam is the most important beam of 
that material with unequal flanges. In the case of wrought 
iron the two coefficients of elasticity are ^ ^^^ g ^-^ ^ 
essentially equal to each other ; conse- 



^C7 

D 



quently the axis about which the moment \37 — \ 

of inertia of the section is to be taken r 
passes through the centre of gravity of the 

latter. ^/^ 

A It 1 . . 1 . , . rig.3 

All the experiments cited in this sec- 
tion are those of Sir William Fairbairn, given in his " Useful 
Information for Engineers," first series. 



56o 



UNEQ UAL-FLA NGED BEAMS. 



[Art. 64. 



Experiment L 

A section of the beam is shown in Fig. 3. It was composed 
of two 2^-inch Ls riveted to a ; x Vx- 

B D 

[^||^;~~ — I inch plate. AD was horizontal, and the 

flange, BF, downward ; hence F was in 
tension. 

W = centre breaking weight = 3,409 
lbs. 

/, by Eq. (29) of Art. 49, = 1.738. 

jTj = distance of centre of gravity from 
F = 1. 91 inches. 



Fig.4 



Span = / = 7 ft. = 84 inches. 

JC = T' = apparent intensity of tensile stress at F, 

Hence: 

W/r 
K= r = -^^ = 78,400 lbs. 

Experiment II. 

Beam and data the same as before, except: 

W — 7,750 lbs. 

/ = 27 inches. 
Hence: 

K=T'-^'^= 57,344 lbs. 

Experiment III. 

Beam and data the same as before, except : 
BF was upward, causing compression at F, ' 

W — 10,777 lbs. 

/ = 27 inches. 



Art. 64.] WROUGHT IRON. 56 1 

K = C = apparent intensity of compressive stress at F. 

Hence: 

K= C = 78,400 lbs. 

Experiments II. and III. were made by testing portions of 
the same beam used in Experiment I. 

Experiment IV. 

A section of the beam is shown in Fig. 4., but it was 
tested with the rib or web upward, as shown in Fig. 2. 

AB = 2.85 inches. BF = 2.5 inches. 

Thickness of rib = 0.29 inch. 

Thickness of flange = 0.375 inch. 

W = 3,019 lbs. / = 48 inches. 

x^ distance of centre of gravity from F = 1.86 inches. 

/= 0.989. 



Hence : 



W/x 
K= C = -^^ = 68,100 lbs. 



Experiment V. 



Beam and data same as for IV., except 
Rib was downward, as shown in Fig. 4 : 

W = 3,153 lbs. 
36 



5^2 UNEQUAL-FLANGED BEAMS. [Art. 64. 

Hence : 

• K = T' = 71,000 lbs. 

In all these experiments half the weight of the beam was 
included in W. * 

These results show that the apparent ultimate intensities of 
resistance to compression and tension in bending of X beams 
may be taken equal to each other; also that there may be 
taken : 

K = C = T' — 70,000 lbs. per sq. in. 

The ultimate tensile resistance (Z) of this iron probably 
ranged from 45,000 to 50,000 pounds per square inch. Hence, 
nearly : 

2 

From the equality of C and T' ^ it follows that the beam is 
equally strong whether the web or rib is up or down. 



Deflection of Wrotight-Iron T Beams. 

If w is the centre deflection of a beam loaded with the 
centre weight W, E the coefficient of transverse elasticity, and 
/ the span, then, as has been seen : 

^-^^^ (.^\ 



or. 






A mean of the experiments II. and III. gave: 



Art. 64.] WROUGHT IRON. 563 

rrr 11 O. I 7 + O 1 8 . - 

W — 4,040 lbs., w = — - — = 0.175 inches. 



Hence: 



/= 1.738. 



Z7 ^^' . ^. 



This IS a small value for E, but is due to the fact that the 
beam was a built one."^ 

A mean of the experiments IV. and V. give: 

Tjr iu 0.135 + 0.17 ^ ^ . r „ 

W — 1,400 lbs., w = — ^^—^ = 0.15025 m., /= 0.989. 

Hence: 

£ = 21,706,000. 

This last value of E is about four times as large as the 
other. Hence the rolled beam would deflect only one-quarter 
as much as the built one. All values of W were within the 
elastic limit. 

These values of E, inserted in Eq. (26), will give the deflec- 
tion for a load W (including five-eighths the weight of the 
beam) at the centre. If Wj^ is the total uniform load, S/^ W^^ is 
to be put for W in the equation. Eq. (26) requires /, w and / 
to be in inches. 

If, however, /Is in feet and other dimensions in inches: 

^ = ^-gj- (28) 

The foregoing formulae, both for breaking weight and deflec- 

* It is probable that the riveting was done by hand. The improved modern 
machine riveting would make a much stififer beam. 



564 



EQUAL-FLANGED BEAMS. 



[Art. 65. 



tion, may be used for the bending of angle irons with sufficient 
accuracy for all ordinary purposes. 

Art. 65. — Flanged Beams with Equal Flanges. 

Nearly all the flanged beams used in engineering practice 
are composed of a web and two equal flanges. It has already 
been seen that the ultimate resistances, T and C, of wrought 
iron, to tension and compression are essentially equal to each 
other; the same may be said also of its coefficients of elastic- 
ity. While these observations may not be applied with pre- 
cisely equal force or truth to the milder forms of steel now 
working their way, to a considerable extent, into engineering 
construction, they certainly hold without essential error. 

In Fig. I is represented the normal cross section of an equal- 
flanged beam. It also represents what may 
be taken as the section of any wrought 
iron or steel I beam. Although the thick- 
ness /' of the flanges of such beams is not 
uniform, such a mean value may be taken 
as will cause the transformed section of 
H j j B--{ Fig. I to be of the same area as the orig- 
inal section. ♦ 

Unless in very exceptional cases where 
local circumstances compel otherwise, the 
beam is always placed with the web ver- 
tical, since the resistance to bending is 
much greater in that position. The neu- 
tral axis HB will then be at half the depth of the beam. Tak- 
ing the dimensions as shown in Fig. i, the moment of inertia 
of the cross section about the axis HB, is : 



c 

[^ &4 >i 

1 



T" 



1 



±- 



D 



/- ^^' - {b - t)h^ 
~ 12 



(I) 



[Art. 65. FORMULA. 565 

while the moment of inertia about CD has the value : 

A = (2) 

With these values of the moment of inertia, the general 
formula, J/ = — - , becomes (remembering that d^ = — or -) : 



j^^^bjfi-{b-jyi (3) 



Or; 



, _ 2t'b^ + hP 

^'^ ~ ^ 6b 



(4) 



C is written for K, since K = T = C. 

Eq. (3) is the only formula of much real value. It will be 
found very useful in making comparisons with the results of a 
simpler formula to be immediately developed. 

Let d^ =z yi {d -{- h). Since t' is small, compared with — , 

the intensity of stress may be considered constant in each 
flange without much error. In such a case the total stress in 
each flange will be : Cbf = Tbt\ and each of those forces will 
act with the lever arm ^^/^ . Hence the moment of resistance 
of both flanges will be : 

Cbf . d, . 

fJlZ 

The moment of inertia of the web will be : — . Conse- 

12 

quently, its moment of resistance will have very nearly the 

value : 

CtJe_ 
6" • 



566 EQUAL-FLANGED BEAMS. [Art. 65. 

The resisting moment of the whole beam will then be : 

M = C{bfd^ + ^) (5) 

A still further approximation is frequently made by writing 
dji for Ji^ ; then if each flange area bt' — /, Eq. (5) takes the 
form ; 

M=Cd,{f+^^ (6) 

Eq. (6) shows that the resistance of the web is equivalent to 
that of one-sixtJi the same amount concentrated in each flange. 

If the web is very thin, so that its resistance may be neg- 
lected : 

M = Cfd, = Cbtd, . ' (;) 

Or: 

^=Cd^ (^^) 

Cases in which these formulae are admissible will be given 
hereafter. It virtually involves the assumption that the web 
is used wholly in resisting the shear, while the flanges resist the 
whole bending and nothing else. In other words, the web is 
assumed to take the place of the neutral surface in the solid 
beam, v/hile the direct resistance to tension and compression 
of the longitudinal fibres of the latter is entirely supplied by 
the flanges. 

Again recapitulating the greatest moments in the more 
commonly occurring cases : 

Cantilever uniformly loaded : 

2 2 



Art. 65.] FORMULA. 567 

Cantilever loaded at the end : 
M= WL 

Beam supported at each end and uniformly loaded : 

8 8 • 

Beam supported at each eiid and loaded at centre : 

Wl 



Beam supported at each end and loaded both uniformly and 
at centre : 

4 \ 2 

In all cases W is the total load or single load, while /, as 
usual, is the intensity of uniform load, and / the length of the 
beam. 

In '' Useful Information for Architects, Engineers and 
Workers in Wrought Iron," issued by the Phoenix Iron Co. of 
Phoenixville, Penn., are the record of some experiments by 
which the value oi C ox T may be determined. These will now 
be used. 

Example I. 

A 7-inch I was subjected to successive loads at the centre 
of the span, the ends being simply supported. The beam 
weighed 60 pounds per yard; consequently the area of the 
cross section was 6 square inches. The span was 21 feet, or 
252 inches. The dimensions represented in Fig. i are the fol- 
lowing: 



568 



{b-t) = 





EYE 


BEAMS. 




t — 0.36 i 


nches. 






h = 5.95 


<< 


.-. /^3 :3= 


210.63. 


d = 7.00 


(( 


.-. ^3 ^ 


343. 


d = 3.67 


(< 






^) = 3-31 


*• 






/' = 0.525 


<< 






d.= 'A (^+ /O = 


: 6.475 inches. 




f=bf 




= 1.927 " 





[Art. 65. 



The following table gives all the recorded results. 



CENTRE 


DEFLEC- 


PERMA- 


REMARKS. 


^=U ('^+H- 


LOAD. 


TION. 


NENT SET. 




Lbs. 


Ins. 


Ins. 




Ins. 


2,000 


0.468 






W = 0.537 


3,000 


0.743 






w = 0.775 


4,000 


1.020 






w = 1. 012 


5,000 


1.298 


0.029 


Weight removed. . . 


7U = 1.250 


6,000 


1.578 


0.030 


it (( 


The coefficient of elasticity, 


7,000 


1.887 


0.060 


(i (< 


£, is taken at 30,000,000 


8,000 


2.300 


0.183 


(( (< 


lbs. 


9,000 


3-540 








9.500 


5.298 








10,000 











With the load of 10,000 pounds at the centre the ** beam 
sunk slowly, top flange yielding." The beam, therefore, may 
be considered as essentially failing with a load of 10,000 pounds 
at its middle point. As the top flange yielded, the ultimate 
resistance to compression, or C, will be given by the experi- 
ment. 

In reality, the beam carried a uniform load of 20 pounds 
per foot (its own weight), besides the single load of 10,000 
pounds at the centre. Hence, Eq. (22) of Art. 24 will give the 
value of M. It is as follows : 



Art. 65.] EXPERIMENTS. 569 

pi 
But^^ = 20 X 21 -7- 2 = 210; J^ — 10,000, and / = 252. 

/ is taken In inches because the dimensions of the cross section 
are in the same unit. These values give : 

M = 643,230. 

Also the data given above, placed in Eq. (3), give : 

M= C X 13.37. 

Equating these values : 

C = 643,230 -r- 13.37 — 48,110.00 pounds . . (8) 

Again, the proper data inserted in Eq. (6), the approximate 
formula, give : 

M — C X 14.79. 
Hence : 

C = 643,230 -f- 14.79 — 43>490'00 pounds . . (9) 

The first permanent set was observed with a centre load of 
5,000.00 pounds. This gives a bending moment at centre of 

J/ = - (^ 5,000 + y) = 328,230. 

Hence : 

C = 328,230 -^ 13.37 = 24,550 pounds. 

As the permanent set with this load was very small, and as 



S70 EYE BEAMS. [Art. 65. 

there was none at all observed with a centre load of 4,000 
pounds (nearly corresponding to (7 = 20,000.00 pounds), the 
limit of elasticity may be taken at about : 

20,000 + 24,000 , 
■ — 22,000.00 pounds. 



In the right hand column of the table are calculated the 
deflections by Eq. (21) of Art. 24, the coefficient of elasticity 
being taken at 30,000,000.00 pounds. By Eq. (i), using the 
data already given : 



Hence : 
Also : 



/ = 46795. 
/3 ~ Af^EI = 0.0002375. 



|//=: 262.5. 



These values inserted in the formula give the results shown in 
the table. The experimental quantities are seen to increase 
much more rapidly than the results given by the formula. The 
agreement, however, is sufficiently close for ordinary purposes. 

Example II, 

The second example, derived from the same source as the 
first, is that of a 9-inch I, 87 pounds per yard. The data to be 
used in connection with Fig. i are as follows : 

/' = 0.72 inches. 

b = 4.00 " .*. f = bt' ^ 2.88. 

/ = 0.39 '' 

d = 9.00 *' .-. d^ = 729.000. 

/i z= 7.56 " .'. //3 = 432.581. 

{b - = 3-6i " 



Art. 65.] 



EXPERIMENTS. 



571 



/ = 21 feet =252 inches ; p — 2^ pounds per foot, 
^i = 8.28 inches. JF = 17,500.00 pounds. 

The bending moment at centre, as before, is : 
AT = ^ Ar + ^) = 1,121,683.5. 

The above data inserted in Eq. (3) give : 

M = C X 25.08. 
Hence : 

C = 1,121,683.5 -r- 25.08 = 44,724.00 pounds. . 
Again the approximate formula Eq. (6) gives : 

M = C X 27.92. 
Hence : 

C= 1,121,683.5 "^ 27.92 = 40,175.00 pounds. . 



(10) 



(") 



The results of this experiment are given in the following 
table, exactly as in Ex. I. 



CENTRE 


DEFLEC- 


PERMA- 


REMARKS. 


--Jei (^+10- 


LOAD. 


TION. 


NENT SET. 




Lbs. 


Ins. 


Ins. 




Ins. 


2,000 


0.228 






0.257 


4,000 


0.474 






0-454 


6,000 


• 0.720 






0.651 


8,000 


0.962 






0.848 


10,000 


I.20T 


0.048 


Weight removed. 


1.045 


12,000 


1.432 


0.050 


t > 1 < 


E is taken at 30,000,000.00 


13,000 


1,580 


0. 117 


1 1 ( ( 


lbs. 


14,000 


1.863 


0.269 


(I 4t 




16,000 


3-256 








17,000 


5-233 








17,500 


5.602 









5/2 EYE BEAMS. [Art. 65. 

The beam may be considered as having yielded, in failure, 
with a centre load of 17,500.00 pounds. That number was 
consequently taken above in the greatest value of M, 

If it be assumed that permanent set was just at the point 
of beginning with the centre load of 9,000.00 pounds, which 
cannot be far wrong, the corresponding moment will be : 

M=~ ("9,000 + 4) = 586,152 

.*. C = 586,152 ~ 25.08 = 23,370.00 pounds (limit of elas.). 

Taking a mean of the results of the two examples : 
By exact formula [Eq. (3)] : 

C = 46,417.00 pounds. 

By app. formula [Eq. (6)]: 

C^ = 41,833.00 pounds. 

For the limit of elasticity : 

C^ = 22,700.00 pounds (nearly). 

These results may be considered accurate for the Phoenix 
Iron Co.'s beams. These experiments were made in 1858. 

It is interesting to notice that these beams failed in the 
compression flanges. 

It is also important to observe that the ultimate resistance, 
Cy is fully equal to the ultimate tensile resistance of good 
wrought iron in large bars. This serves to confirm the opin- 
ion that the ultimate tensile and compressive resistances of 
wrought iron are not far, at most, from being equal to each 
other, and that these quantities may be used for C or Kin the 



Art. 65]. EXPERIMENTS. 573- 

formulae for flanged beams. If the approximate formula, Eq. 
(6), is used, however, according to these results C or K should 
be taken about 0.90 (nine-tenths) of the value used in the 
exact formula, Eq. (3). 

The last column of the second table is calculated by the 
formula, as shown, taking E at 30,000,000.00 pounds. The 
same general observations apply to these results as in the 
preceding example. 



Example III. 

The data for this example are taken from the hand-book 
for 1881 published by the N. J. Steel and Iron Co., Trenton, 
N. J., where the beams were broken. The breaking weight is 
the mean of two results for light 6-inch wrought iron Is. 

d = 6.00 ins. t = 0.25 in. f — 0.456 in. 

/ = 12 ft. = 144 ins. / = 23.815, by Eq. (i). 

Since the beam weighed 40 pounds per yard : 
W = 14,000 -f 80 = 14,080 lbs. (centre breaking load). 
Hence : 

C — — J — 63,840 lbs. per square inch. 

By approximate formula : 

J = 0.21. /= 1.368 .-. %+f= 1-578. 

d^ = 5.544 ins. M = 506,880. 



574 EYE BEAMS. [Art. 65. 

Hence, by Eq. (6) : 

(7, = 57>930 lbs. per square inch. 

Example IV, 

A 9-inch heavy Trenton beam, 85 pounds per yard. The 
data are taken from the same source as were those in Ex. 
III. 

^ = 9.00 ins. / = 0.38 in. /' = 0.68 in. 

/ = 15 ft. = 180 ins. / = 108.47, by Eq. (i). 
W = 32,000 4" 212 = 32,212 lbs. (al centre). 
Hence : 

C — — Y — 60,120 lbs. per square inch. 

By approximate formula : 

— ^ 0.484. f= 2.72 .-. — +/= 3.204. 

d^ := 8.32 ins. M = 1,449,540. 

Hence by Eq. (6) : 

C — 54>370 lbs. per square inch. 



Taking the means of these two sets of results : 



Art. 65.] U. S. TEST BOARD'S EXPERIMENTS, 575 



By exact formula [Eq. (3)] : 

C = 61,980. 

By app. formula [Eq. (6)] : 

Q = 56, 1 so- 
All the conclusions reached in connection with Exs. I. and 
II. are confirmed by the results of Exs. III. and IV. 

C and Cj^ are much larger, however, for the Trenton than for 
the Phoenix beams, and both are very high for beams of such 
length of span with no lateral support for the compression 
flange. 

In calculating the deflection of rolled wrought-iron beams 
E may be taken from 28,000,000 to 30,000,000. 

The exact formulae of this Article are strictly applicable to 
rolled beams only, but the approximate formula finds exten- 
sive application in cases of built beams. 

Experiments by U, S, Test Board, 

Table I. contains the results of a valuable series of tests by 
the U. S. Board, ** Ex. Doc. 23, House of Rep., 46th Congress, 
2d Session." 

The values of ^and E at elastic limit are computed from 
data contained in that document in the manner already shown 
in detail, and which it is not necessary to repeat. It is both 
interesting and important to observe the considerable, though 
irregular, increase of the intensity of stress in the exterior fibre, 
at the elastic limit, with the decrease of depth. E is seen to 
vary from 26,099,400 to 36,664,400, with a mean value of 
31,128,260. As a general result, E is slightly larger for the 
smaller beams than for the larger. 



5/6 



EYE BEAMS. 



[Art. 65. 






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Art. 65.] 



TESTS OF COL. LA IDLE Y. 



S77 



A chemical analysis of six specimens from these beams 
gave the following results. 

These experiments were conducted by Gen'l Wm. Sooy 
Smith, who kindly gave to the writer the final centre loads and 
deflections. 



PERCENTAGES OF 



Sulphur. 


Phosph'us. 


Silicon. 


Total Carb. 


Manganese. 


Copper. 


Cobalt. 


Nickel. 


O.OIO 


0.436 


0.189 


0.031 


0.031 


0.012 


0.029 


0.029 


0.008 


0.447 


0.190 


0.038 


0.028 


0.008 


0.021 


0.023 


O.OIO 


0-453 


0.203 


0.037 


0.028 


O.OIO 


0.015 


0.018 


0.012 


0.423 


0.182 


0.039 


0.022 


0.022 


O.OIO 


0.015 


0.005 


0.271 


0.177 


0.027 


0.028 


0.052 


0.031 


0.026 


O.OII 


0-375 


0.197 


0.039 


0.028 


0.010 


0.018 


0.016 



Col. T. T. S. Laidley, U. S. A., has also completed the 
tests of a few beams to failure, the results of which are given 
in "Ex. Doc. 12, 47th Congress, ist Session." Table II. con- 
tains values of K at failure, computed from Col. Laidley*s 
data. Beams No. i carried a load uniform from end to end by 

TABLE II. 





DEPTH, 


SPAN IN 






LATERAL SUP- 


TOTAL LOAD 


A' IN LBS. 


NO. 






MEAN OF 


LOAD. 










INS. 


FEET. 


• 




PORT. 


IN POUNDS. 


PER SQ. IN. 


I 


I5-0 


28.5 


3 Exps. 


Uniform. 


Uniform. 


118,000 


54,260 


2 


I5-0 


28.0 


2 Exps. 


Centre. 


None. 


28,650 


25,890 


3 


10.5 


17.0 


2 Exps. 


Centre. 


None. 


21,020 


32,210 


4 


10.5 


17.0 


I Exp. 


Centre. 


None. 


22,020 


38,070 



37 



57^" BUILT BEAMS. [Art. 66. 

means of brick masonry arches, which thus also gave to them 
a uniform lateral support. This lateral support produced a 
very high value of K^ i.e., 54,260 pounds, which fell to 25,890 
with no lateral support. In the latter case nothing prevented 
the compression flange yielding laterally like a column. The 
10.5-inch beams were much shorter, and the long column influ- 
ence less marked ; consequently the values of K are correspond- 
ingly higher. The tests are not sufficiently numerous to fix 
the law of the decrease of i^with the increase of span. 

Beams Nos. i and 2 weighed 200 pounds per yard, with a 
moment of inertia (/) equal to 706.6. Beam No. 3 weighed 
105 pounds per yard, and gave /= 174.75; while No. 4 
weighed 92 pounds per yard, with / = i54-9' 



Art. 66. — Built Flange Beams with Equal Flanges.— Cover Plates. 

A "built beam" is a beam built up of plates and angles 
like that shown in Fig. i. As shown in that figure the web is 
composed of a single plate, called the ''web plate," supported 
by " stiffeners," if necessary, as is usually the case. These stiff- 
eners are vertical pieces of Ls or J_s riveted to the web plate, 
in accordance with principles to be shown hereafter. The 
flanges, as shown by the heavy lines, are composed of Ls and 
plates so arranged as to give the requisite area of cross section 
at any point. 

The method of designing such a beam, and the calculation 
of the elements of its resistance, will be given in detail. The 
beam is supposed to be of wrought iron, and one of a system 
for a double track railway bridge ; the stringers under the two 
tracks, which rest on the beam, are placed at A and B, and D 
and //. The weight of the beam, taken uniformly distributed, 
IS 5,300.00 pounds. The concentrated load at each of the 
points A, B, D and //, composed of the train weight added to 
that oi the stringers, is 42,000.00 pounds. 



Art. 66,] 



DATA. 



579 



The following are some of the dimensions of the beam : 

Span RR' = 26.5 feet. Depth of web plate = 36 inches. 
R'II=RA = 3.25 feet. nH = AB = 6.00 feet. BB = 8.00 
feet. 

The web plate will be taken j\ inch thick. The method of 
determining this thickness will be shown hereafter. 

In this case resistance to flexure of the web will be neg- 
lected ; ^/le web will be assumed to resist the shear 07tly, as is 
assumed in Eqs. (7) and (/a) of Art. 65. The ** depth," d^, of the 




Fig.1 



beam will then be the vertical distance between the centres of 
gravity of the sections of the flanges, and each flange is to be 
considered as composed of two Ls and the ** cover'* plate or 
plates only ; no part of the web is to be included. Strictly 
speaking, then, the depth is variable ; but this variation is so 
slight that no essential error will be committed if it be con- 
sidered constant and equal to the depth of web plate, or 36 
inches. This procedure, which saves much labor and time, is 
always permissible where cover plates are used. The next ex- 
ample will exhibit a case in which they are not used. 

TJie direct stresses of tension and compression existing in the 
flanges must be carried through the rivets which unite the flanges 
to the zueb ; hence the necessary number of those rivets will 
first be determined. 

The reaction at R, using the data already given, will be : 



R = 2 X 42,000.00 -f- — — '- — = 86,650.00 pounds. 



5 So BUILT BEAMS. [Art. 66, 

The weight per lineal foot of floor beam is : 

5,300.00 , 

:^^-^ = 200.00 pounds = w. 

26.S ^ 

The bending moments for the two sections A and By will 
next be found. 

Moment at 

A = (86,650 — 100 X 3.25) 3.25 = 280,500.00 nearly. 

Moment at 

B = 86,650 X 9.25 — 42,000 X 6 — 100 (9.25)^ = 463,950.00. 

Since the depth of the beam is 3 feet : 
Flange stress at 

A = 280,500.00 -^ 3 = 93,500.00 pounds. 
Flange stress at 

B = 463,950.00 -f- 3 = 154,650.00 pounds. 

The allowable intensity of pressure between the rivet and 
its hole (see Art. 73) will be taken at 10,000.00 pounds. The 
diameter of rivets is a matter of judgment ; it will be taken at 
^-inch. Rivets for built beams usually range from ^ to i 
inch in diameter. 

The selection of the Ls for the flanges is also, to some 
extent, a matter of judgment. In the present instance, 
4" X 4"Ls, 50 pounds per yard, will be taken. These will be 
found to answer the purpose. 



Art. 66.] RIVETS AND FLANGE STRESSES. 58 1 

The effective bearing surface between each rivet and the 
web plate will then be : 

7 7 

-^ X -^ = 0.383 square inch. 

Hence each rivet may carry: 

0.383 X 10,000.00 = 3,830.00 pounds. 

Consequently the number of rivets between R and A should be: 

93,500.00 -f- 3,830.00 = 24 (nearly). 

The increase of flange stress between A and B is: 
154,650 — 93,500 = 61,150.00 pounds. 

Hence the number of rivets required between A and B is : 

61,150.00 -i- 3,830.00 = 16 (nearly). 

Since 24 rivets are required between R and A^ the corre- 
sponding pitch would be but a little more than one and one- 
half inches, which is very much too small. With a ^-inch 
rivet, a three-inch pitch is about the least advisable. If the 
rivets be placed at a pitch of three inches between R and B, 
thirty-seven will thus be located. Three or four rivets, if nec- 
essary, may be put in at the end with a pitch of two inches 
without harm. Again, plates at the upper corners of the beam^ 
as shown, carrying at least six rivets each, should be put on. 
In such methods as these, nearly the full number of rivets 
required between R and A may be supplied, while the two or 
three lacking will be found, without danger to the bbem, adja- 
cent to A on the side towards B. Three or four in excess of 
the number required will be found between A and B, 



582 BUILT BEAMS. [Art. 66. 

No central bending moment at C has been computed, be- 
cause the only difference between such a one and that at either 
B or D is due to the weight of the beam only. This difference 
is essentially nothing. The proper support of the Ls in com- 
pression, however, requires that the rivets be pitched at about 
six inches between B and D, In ordinary floor beams a proper 
bond between the flanges and web requires that the pitch should 
never be greater than about' six or eight inches. 

The shearing of the rivets is not considered, because they 
sustain double shear in the flanges, and their bearing capacity 
is by far the least of the two. 

The rivets, of course, should be pitched alike in both top 
and bottom flanges. • 

The greatest allowable intensity of tensile stress in the bot- 
tom flange will be taken at 8,000.00 pounds per square inch, 
and an equal intensity will be taken for the compressive stress 
in the upper flange. The area required in the bottom flange 
at -^ is : 

~^ — • = 1 1.7 sq. ms. (nearly). 
8,000 ^ ^ 



That required at ^ is : 
154,650 



8,000 



= 19.3 sq. ins. (nearly). 



The area of the two 4" X 4" Ls, 50 pounds per yard, is 10.00 
square inches. The thickness of the angle iron where it is 
pierced by the rivets binding it to the web is about 0.6 inch. 
Hence the area of metal taken out by one rivet is : 

0.875 X 0.6 X 2 = 1.05 sq. ins. 

Or, the effective area of the Ls at A is : 

10.00 — 1.05 = 8.95 square inches. 



Art. 66.] FLANGES. 5^3 

Now, since the weight of the beam itself is small, compared 
with the weight of the train, the flange stress, or moment, 
varies almost uniformly from R to A, Hence, an increased 
section is first needed at 

(8.95 -T- 1 1.7) X 3.25 = 2.5 feet (nearly), 

from R. Since, however, the cover plate to be added must 
take its stress through the rivets which bind it to the Ls, it 
should overlap the necessary distance by one and a half to 
twice its width. In the present case, then, instead of begin- 
ning the cover plate at just 2.5 feet from R, a 12" x j\" cover 
plate will begin at 9 inches from R and extend along the beam 
to a point at the same distance from R'. The length of this 
cover plate will then be 26.5 — 1.5 = 25 feet. This cover plate 
will be bound to the angle irons by ^" rivets, which should, so 
far as possible, be pitched half way between the ^" rivets in 
the other legs of the angle irons. The effective area of this 
cover plate, for tensile stress, will then be : 

(12 — 240.75) X A =" 5-9 sq. ins. (nearly). 

The available area of two Ls and one cover plate is, conse- 
quently : 

8.95 + 5-9 = I4-S5 sq. ins. 

For the reason already given, the moment, or flange stress, 
varies nearly uniformly between yi and B,hut at a different rate 
than between R and A. Since AB is 6.00 feet, the point at 
which another increase of section must begin is at the distance 

[(14.85 — 11.7) -^- (19.3 — 1 1.7)] X 6.00 = 2.5 feet (nearly) 

from A. Again, as in the previous instance, a second cover 
plate, 12" X }4", will be put on, and it will begin, not at 2.5 



584 BUILT BEAMS. [Art. 66, 

feet from A, but at one foot from that point. The available" 
area of this plate will be : 

(12 - 1.5) X >^ = 5.25 sq. ins. 

The total area at the centre of the beam available for ten- 
sion will then be: 

14.85 + 5.25 — 20.10 sq. ins. 

As 19.3 square inches, only, is required at ^, the total effect- 
ive flange area of 20.10 square inches will be sufficient. The 
tension flange has been considered, but the same design will 
evidently give a sufficiently strong compression flange. This 
arises from two causes. In the first place, it has already been 
seen that the ultimate tensile and compressive resistances of 
wrought iron may be taken essentially the same. Again, in 
first class riveted work the rivets so thoroughly fill the holes 
that the metal taken out by the punch or drill need not be 
deducted ; in other words, the effective area is equal to the 
total area of flange section. 

The number of rivets required in a cover plate is yet an 
important question. Since all stress carried by the cover plates 
must be given to them by the rivets, the raouber of rivets be- 
tween the end of any cover plate and that point at which a fur- 
ther increase of flange section is necessary, must be sufficient to 
carry all the stress in the cover plate itself. 

Applying this principle to the first cover plate found neces- 
sary : The load which each J" rivet in the 12" x yV' cover may 
carry is : 

0-75 X tV X 10,000.00 = 4,220.00 pounds. 

The total tensile stress carried by the 12" x yV ' cover is: 
5.9 X 8,000 = 53,200.00 pounds. Hence the number of rivets 
required is : 



Art. 66.] FLANGES. 585 

53,200 -4- 4,220 = 13 (nearly). 

According to the design it is 5 feet from the end of this 
cover to a point 2.5 feet from A toward B, where the next in- 
crease in section is required ; and over this 5 feet these 13 
rivets must be distributed. But in order that a proper bond 
between the component parts of the flange may be obtained, it 
it seldom advisable to make the pitch over 6", and at the end 
of the cover plate this pitch should be halved for about three 



000 o o o o o o o o'\ 

ooop O O O O O O Oi) 

5 1 i-_i 



Flg.2 



rivets. Proceeding in this manner, that part of the bottom of 
the beam, at the end nearest R in Fig. i, which includes the 5 
feet of cover under consideration, will present the appearance 
of the sketch in Fig. 2. RG is 0.75 foot and GF 5 feet. In 
this manner 22 rivets are introduced instead of 13, but it is 
advisable to put in the extra number. 

In the compression flange other considerations appear be- 
sides the simple bearing capacity of the shaft of the rivet. 
Between any two consecutive rivets the cover plate forms a solid 
rectangular column with essejitially fixed ends, whose length is the 
pitch of rivets. The pitch, therefore, must not be sufficiently 
great to^ allow the existence of any material amount of long 
column flexure. Unless plates, therefore, are very heavy, the 
greatest pitch should not exceed about six or eight inches. 

The bearing capacity of a J" rivet in 12" x V' cover is : 

0.75 X 0.5 X 10,000.00 = 3,750.00 pounds. 
The full tensile capacity of the cover plate is : 



586 BUILT BEAMS. [Art. 66, 

5.25 X 8,000.00 — 42,000.00 pounds. 
Hence, the number of rivets required is : 

42,000.00 ~ 3,750.00 = 1 1 (nearly). 

The end of the cover plate, as designed, is one foot from A 
towards B ; and the eleven rivets are nearly all required be- 
tween that end and B, a distance of 5 feet. Hence, if the 
rivets are pitched in this cover plate, near the ends, as shown 
in Fig. 2 for the other cover, and at six inches over the inter- 
vening space, more than the number just determined will be 
introduced. For the reasons already given, however, the num- 
ber will really be not too great. 

In each flange, then, there will be found the following 
pieces properly joined : 

2 — 4" X 4" Ls, 50 pounds per yard. 
I— 12" X yV' plate, 25 feet long. 
I — 12" X i" plate, 18 feet long. 

At the ends of the beam R and R ^ Fig. i, provision must 
be made for the reaction. In this example the reaction is 
86,650.00 pounds. The transverse shearing resistance of the 
web should at least equal this at the ends. The area of a trans- 
verse section of the. web is : 

36 X i^ = 15.75 sq. ins. 

If the greatest allowable shearing intensity in the web be 
taken at 8,000.00 pounds, its shearing resistance will be : 

15-75 X 8,000.00 = 126,000.00 pounds. 
This result is about 50 per cent, greater than is required. 



Art. 66.] ENDS. 587 

Hence safety, so far as shearing is concerned, is amply secured. 
But the end of the beam is also subject to an upward pressure 
of 86,650.00 pounds, which must also be provided for. Two 
6" X 4" X 0" Ls will be riveted to the ends as shown in Fig. 
I, one on each side of the web, and the 6" legs lying against 
it. By pitching -J" rivets at 3" (nearly), in a zigzag manner, 
20 rivets can be introduced to hold these 4" x 6" x 4" Ls to 
the web. The carrying capacity of each f " rivet against the 
web plate has already been found to be 3,830.00 pounds. 
These 20 rivets therefore will carry 3,830 x 20 = 76,600.00 
pounds. Since the area of the cross section of two 4" X 6" X 
|-" Ls is about 10 sq. ins., the bearing of the rivets against the 
web plate is all that need be considered in this connection. 

A proper bearing for the difference 86,650 — 76,600 = 
10,050.00 pounds remains to be found. If it be supposed that 
the ends of the beam rest on ''shoes" or brackets, or other 
supports, it is only necessary that at least 2\ inches of the edge 
of the web plate bear on such support ; for the bearing surface 
will be 2\ X yV = 1. 00 sq. in. (nearly), and it will carry 
10,050.00 pounds. In all ordinary cases much more than that 
amount of edge will bear on the support. It is to be remem- 
bered that in such an instance as this, tJie loiver ends of the 4" 
X 6" Ls must bear fairly and truly against the angle iro7ts com- 
posing the lozver flange, in order that they may take up their 
proper amount of the reaction. 

In some cases the ends of the beam are to be secured to 
vertical surfaces without any supporting shoe or bracket. The 
entire reaction of such a beam must be carried by the vertical 
angles at the ends. The number of ^-" rivets required to hold 
these angle irons to the web would then be 86,650 -^ 3*830 = 
23 (nearly). By shortening the pitch a little these could easily 
be worked into the longer legs of the 6" X 4" X J" Ls- 12 
rivets would then be put through each of the two 4" legs and 
the vertical surface to which the beam is secured. 

No account has heretofore been taken of the shearing re- 



588 BUILT BEAMS. [Art. 66, 

sistance of the rivets, because that has been much greater than 
their bearing capacity, but instances may occur in which such 
a condition of things does not exist. Hence the shearing 
and bearing capacities should always be estimated, and se- 
curity taken in reference to that which is least. As an ex- 
ample : at 8,000.00 pounds per sq. in. the shearing resistance 
of a I" rivet is (0.875)^ X 0.7854 X 8,000.00 — 4,800.00 pounds 
(nearly) ; while the bearing capacity of the same rivet in the 
6" X 4" X i" L is only : 

0.875 X 0-5 X 10,000.00 = 4,375.00 pounds. 

Precisely the same operations are required in determining 
the number of rivets in the vertical Ls at A and B, Fig. i, as 
in those at the ends of the beam ; consequently it is not neces- 
sary to repeat them. 

Thus, there is completed the operation of designing the 
beam, with the exception of finding the thickness of the web, 
which will be given hereafter. 

In general two or three things are to be observed. The 
number of rivets actually required by these calculations should al- 
ways be, as they Just have been, somewhat exceeded. In the best 
of riveted work the rivets will not exactly fill the holes, and 
the beam will not act perfectly as one continuous whole. 

Again, stress is given to the flanges along the line of the 
rivet holes, which is- some distance from the centre of gravity 
of the cross section of the flange. Consequently, some bend- 
ing will be induced in both flanges, and this necessitates some 
extra material. This excess may be estimated if desirable, but 
ordinarily it is entirely unnecessary. The existence of this 
bending demonstrates the advisability of putting on as few 
cover plates as possible. It is far better to use heavier Ls with 
a little waste of material at the ends. 

It is also better to use one heavy cover plate than two thin 
ones having an equal combined thickness, even though the use 



Art. 6J^, 



DATA. 



589 



of the former entails a little waste ; for the heavy plate be- 
tween two consecutive rivets will resist far more bending as a 
column than the two others each of half the thickness. 

If the end of the beam were made as shown in Fig. 3, no 
web plate would be re- 
quired between R and 
Aj for all shear would 
be carried by the in- 
clined flange. 

The upper flange, 
being in compression, 

would require riveting, but none would be needed in the 
lower, except in the immediate vicinity of R. The flange 
stresses between A and R would also be uniform, instead of 
uniformly varying as in Fig. i . 




Fig.3 



Art. 67. — Built Flanged Beams with Equal Flanges. — No Cover Plates. 

The flanged beam represented in Fig. i is supposed to carry 
a portion of the floor of a highway bridge. In this case, also, 
the bending resistance of the web plate will be neglected. The 
beam proper is the portion RR' R'R, supported at RR and 
R'R' ; while the portions ARR and HR'R' form cantilevers for 
the support of the sidewalks. 

The following are the dimensions : 

AR = HR' = 6 feet. RR' = 28 feet. 

AH = 40 feet. RR = R'R' = 31 inches. 

RB = BM = MF = FR' = 7 feet. 



The depth RR has been taken at 31 inches, so that the 
effective depth to be used in finding the flange stresses will 
be about 2.5 feet. 



590 



BUILT BEAMS. 



[Art. 6j, 



The weight of the beam proper, RR'R'R, added to the 
flooring which it supports, is taken at 



at 



14,650.00 pounds. 



The greatest uniform load between R and R' will be taken 



37,440.00 pounds. 



M 



R 




Fig.l 




R' 



Hence the total uniform load to which the beam is sub- 
jected is : 

37,440.00 + 14,650.00 — 52,090.00 pounds. 

The weight of one cantilever, with the flooring which it sup- 
ports, will be taken at 

3,100.00 pounds. 
The total moving load on AR, or HR' , will be taken at 

8,640.00 pounds. 
The total load, therefore, carried by one cantilever is : 
3,100.00 4- 8,640.00 = 11,740.00 pounds. 



The beam proper, RR\ may sustain its greatest load when 



Art. 67.] NO COVER PLATES. 59 1 

the sidewalks carry nothing but their fixed weight. This con- 
dition of things will cause the greatest compression in the 
upper flange and tension in the lower, and will be assumed in 
designing the beam. 

The fixed weight of a cantilever will cause stresses in the 
flanges of opposite kinds to those produced in the beam, but 
of such small amount that they will be neglected ; the neglect 
originating a very small error on the side of safety. 

The total load per linear foot of RR is : 

52,090.00 -^ 28 = 1,860.00 pounds. 

The flange stress in the beam at R will be nothing ; it will 
be found at the two points B and M. Strictly, the *' depth " 
to be used should be the vertical distance between the centres 
of gravity of the flanges. It will not be far wrong to take this 
depth at 2.5 feet, since the web plate is 31 inches deep. The 
reaction at i^ is : 

52,090.00 -4- 2 == 26,045.00 pounds. 
The flange stress at ^ is : 
(26,045 X 7 — 1,860 X {jY -^ 2) -t- 2.5 = 54,700.00 pounds. 
The flange stress at the centre M is : 

(52,090 X 28 -T- 8) -^ 2. 5 = 72,926.00 pounds. 

If, as in the preceding Article, the greatest allowable stress 
in the flanges is 8,000.00 pounds per square inch, a flange area 
of 9.1 15 square inches is required in the present case. If each 
flange is composed of 2 — 4" x 6" x i" Ls, 50 pounds per 
yard, there will be a very little excess of flange area, as there 



592 BUILT BEAMS. [Art. 6/. 

should be ; these Ls will then be taken for the flange, the 4" 
legs being riveted to the web plate ; \" rivets will be used in 
riveting the flanges to the web. Where pierced by the rivets, 
the legs of the Ls are about ^' thick. Hence a rivet hole will 
cut out 2 X J X 0.875 = 0.875 square inch. There will then 
still remain lo.o — 0.875 = 9-125 square inches of effective 
area, which is a little in excess of the 9. 115 required. 

A web plate |-" thick will be assumed. Taking 10,000.00 
pounds per square inch as the greatest allowable intensity of 
pressure between shaft of rivet and plate, the bearing capacity 
of each rivet will be : 

0.875 X 0.375 X 10,000 = 3,280.00 pounds. 

In this case all the moving load rests upon the top of the 
beam, and since the edge of the web plate is only 0.375" wide, 
that moving load must be taken as resting on the Ls of the 
upper flange, and hence indirectly on the rivets. Also, since 
nearly the whole of the fixed load rests upon the upper flange, 
the entire load of tJie beaut will be taken as resting on that flange. 
Consequently, between R and B the rivets will be subjected to 
the action of a vertical force equal to 1,860 X 7 = 13,020.00 
pounds, and a horizontal one equal to 54,700.00 pounds. The 
resultant force will then be : 



'v/(i3,02oy + (54,700)^ — 56,230.00 pounds. 

Between B and M the vertical force will then be the same, 
but the horizontal one will be 

72,926.00 — 54,700.00 = 18,226.00 pounds. 

The resultant, therefore, is : 



VCi 3,020)^ 4- (18,226)' = 22,400 pounds. 



Art. 6/.] NO COVER PLATES. 593 

Hence the number of rivets required between R and B is : 

56,230.00 -r- 3,280.00 =18 (nearly). 

The number between B and M is : 

22,400.00 ~ 3,280.00 = 7 (nearly). 

If, therefore, commencing at R or R , the rivets be pitched 
at 3 inches for a distance of 4.5 feet, then at 6 inches to the 
centre M^ about 36 or 37 rivets will be found in each hklf of 
each flange. This number is in excess of that required, but 
for the reasons given in the preceding Article, it is probably 
not too many. Thus the flanges are designed without the use 
of cover plates. 

In this case the beam will be suspended from hanger loops 
at R and R! , which carry resting plates or shoes for the beam 
at their lower extremities. 

The total reactions at the lower R and R' will be half the 
total weight of the entire beam with the moving load, or : 

Reaction = (52,090.00 + 23,480.00) ~ 2 — 37,785.00 pounds. 

At i^ and R 2—4" X 4" X V' Ls will be riveted to the beam 
as shown. The' lower ends of these angles should abut firmly 
and squarely against the angles of the lower flange. 

Since the greatest allowable pressure between a rivet and 
the web plate is 3,280 pounds, the number of rivets required 
at each end of the beam in each pair gf vertical Ls is : 

37,785.00 -J- 3,280.00 = 12 (nearly). 

If, consequently, these rivets be pitched at 3 inches, a 
sufificient number will be obtained, if it be remembered that a 



594 BUILT BEAMS. [Art. 6/. 

portion of the edge of the web will bear against the resting 
plate. 

The pitch in the stiffners (4" X 4" X ^V" Ls) at C and D 
may be taken at 6" with an extra rivet at each end. 

The horizontal flange stress for the cantilevers at R and R 
is : 

(11,740.00 X 3) -^ 2.5 = 14,088.00 pounds. 

The secant of the angle which the inclined flange makes 
with the horizontal is about 1.05. Hence the inclined flange 
stress is : 

14,088 X 1.05 == 14,800.00 pounds. 

Hence, if in each flange at A there are 

14,800.00 ^ 3,280 = 5 (nearly) 

rivets, securing the flanges to the piece of plate shown, ample 
security will be obtained. 

The cantilever flanges possess a large excess of material. 

Calculations on the shearing of the rivets between the web 
and flange have not been made, because the resistance of a 
rivet to double shear is much in excess of its bearing capacity. 

The excess of material in the Ls of the ftanges is not as 
much as it really should be, because the line of horizontal 
stress along the rivet holes is 'somewhat below or above the 
centre of gravity of the flange, and some bending is conse- 
quently induced. This bending, however, is not as great as if 
cover plates had been used, and the neglect of the bending 
resistance of the web plate is somewhat of an offset. Besides, 
as has "already been stated, in this particular case, the fixed 
weight of the cantilevers relieves a little of the flange stress of 
the beam as actually found. 

Since the transverse section of the web plate has an area of 



Art. 6S.] 



BOX BEAMS. 



595 



0-375 X 30 = 11.25 square inches, transverse shearing at the 
points of support is more than provided for. 

If either a railway or highway floor beam has a variable 
depth, the operations are in no manner changed. The depth, 
however, to be used in finding the flange stress at any point 
must be the vertical depth at that point. The stress thus 
determined must be multiplied by the secant of the inclination 
to the horizontal at the same point for the inclined flange. 

Art. 68. — Box Beams. 

The class of beams known as box beams in engineering 
practice are represented in Figs, i and 2. In Fig. i the upper 
and lower flanges are each composed of a plate whose thick- 
ness is f and two Ls whose lengths of legs and thickness are 
s and a, respectively. If it be assumed that the web plates. 



-«/- 



JJ 



< w ^ 



< .V ^ 



Fig.l 






T 
I 

d 
I 



±-^ 



•-^ 



£ W i 






Fig.2 



1 - \y 1^ 1 



'T 



,;._A_. 



-I, 



r 



I 

I B 



Fig.3 



the thickness of each of which is /, offer no resistance to bend- 
ing, then the effective depth of the box beam will be the ver- 
tical distance between the centres of gravity of the flanges. 
If / is the area of one of these flanges, and d^ this effective 
depth, the resisting moment of the beam, as has already been 
shown, will be : 



M = CM ; 



(I) 



59^ FORMULA FOR BUILT BEAMS. [Art. 69. 

in which C = K — intensity of stress at the distance Yzd^ from 
the neutral surface. If the flange area is desired: 

^=Cd. (^) 

In other words, the methods and all the operations regard- 
ing rivets, etc., as well as the values of C and T, or K, are 
precisely the same for the box beams as for the other built 
beams of the preceding Articles. 

If each flange is composed of several plates and 4 Ls (as 
shown by one in dotted lines), then /' is to be taken as the 
combined thickness of all the plates, while f will be the com- 
bined area of the several plates and 4 Ls. 

Fig. 2 shows a box beam composed of two channels and 
one or more plates in each flange. The general observ^ations 
applied to Fig. i apply with equal force to Fig. 2. The bend- 
ing resistance of the webs of the channels may be neglected if 
very thin, or when desired in any case, but the exact formula 
to be given in the 'next Article is well adapted to this beam. 



Art. 69. — Exact Formulae for Built Beams. 

The exact formulae for the built sections already given are 
simply the special forms of the general formula : 

M=?L ■..(,) 

The moment of inertia, /, is to be taken about a horizontal 
line through the centre of gravity of the normal section, i.e., 
about a line parallel to the side b in the three Figs, of the pre- 
ceding Article. 



Art. 69.] 



FORMULAE. 



597 



In Fig. I of that Article the moment of inertia of the cross 
section about AB is: 



/ = ^' . bt' ('^ + ^'y + (^ + 'y' 

6 26 



'{s — a) (d — 2a)^ 4- a(d — 2sf' 



. ■ (2) 



If there are four Ls in each flange, one only of which is 
shown in dotted lines: 



_ bt'^ , .. {d + tj {2s + t)d^ 
I_^^bt — ^— + ^ 



'{s — a) {d — 2af + a{d — 2^)3" 



• ■ (3) 



The moment of inertia of the cross section shown in Fig. 2, 
about ABy is : 

j^bj^ ^/' (^ + ty I (^ + t)d^ - s{d - 2df 

6 2 6 • v4J 



The moment of inertia of the cross section shown in Fig. 
3, about AB^ can be either directly written from an examina- 
tion of that Fig., or derived from Eq. (2) by simply writing 

— for /. It has the value : 



/=?Cy + (^+oO + ^-^±#^^ 



\s — d){d — 2df + a{d — 2sy\ , . 

-^ 6 J • • • ^5) 



59^ BUILT BEAMS. [Art. 70. 

,If the plates are omitted from the flanges in Fig. 3, as in 
the Article on built beams without cover plates, t' = o, and 



_ {s + yit)d'^ _ 
6 



'(s — a) {d — 2<^)3 -}- a(d — 2sY 



■ (6) 



In all these cases d^ =^ y2d -\- t' or y^d, according as there 
are or are not cover plates. 

These several values of / and d^y substituted in Eq. (i), will 
give the resisting moments for the various sections. It is an 
open question, however, what degree of accuracy may be ex- 
pected to result in the application of these formulae. It is to 
be remembered that the very best of riveted work does not se- 
cure that degree of continuity presupposed by the Eq. (i). It 
may be stated, however, that Eq. (4) is better applicable to its 
cross section than the others, for there is perfect continuity be- 
tween the web and a part of the flange. 



Art. 70. — Examples of Built Beams Broken by Centre Weight. 
Example I. — Wrought Iron Beam. 

This beam was tested by Sir William Fairbairn (" Useful 
Information for Engineers," first series), and was composed of 
four 2-inch Ls riveted to a 7 by ^-inch web plate. The dis- 
tance between supports was 7 feet or 84 inches. 

A section of the beam is shown by the section, only, of Fig. 
I in Art. Oy ; there were no cover plates. 

The Ls in the bottom flange were a very little heavier than 
those in the upper, but the difference was so small that it has 
been neglected ; or, rather, the small excess has been assumed 
to supply the loss caused by the rivet holes. 

Centre breaking weight = 24,380 -j- 80 = 24,460 lbs. 



Art. 70.] BREAKING TESTS, 599 

Wl 

1=7 feet = 84 inches. .*. M = = 513,660. 

4 

Referring to Eq. (6) of Art. 65 : 

d = 6.5 ins. / = 2.083. -^ = 0-30* 

^ 513,660 ,, 

.*. C = =^—^ ■ = 33,140 lbs. per sq. in. 

</iWas taken as the depth (nearly) between the roots of the Ls. 

The beam gave way in the top, or compression, flange by 
the twisting of the Ls at a comparatively low compressive in- 
tensity. This indicates that the discontinuous riveted connec- 
tion between the web and flange, although the pitch of the 
rivets was only 4.5 inches, fails to give such perfect support to 
the top flange, as a column, as the perfect continuity of the 
connection in a rolled beam. 

The condition of the top flange, as a column, in a built 
beam, therefore, exercises a very important influence on the 
ultimate resistance of the beam, and should not be neglected. 

It is probable, however, that the high compressive resist- 
ance of American wrought iron of the present day would give 
a much higher value of C under the same circumstances. 

When the centre load added to five-eighths the weight of 
the beam was 8,4og pounds, the centre deflection, or Wy was 
0.18 inch. Hence the coefficient of elasticity was: 

±, = ^ a r ~ 12,321,000 pounds. 

482x^7 

/3 must be taken in inches. / was computed by Fairbairn 
at 46.77. 



600 BUILT BEAMS. [Art. 70. 

Example II. — Steel Beam, 

The data for this beam were given by Albert F. Hill, C. E., 
in " Steel in Construction," Engineers' Soc. of West. Penn., 
April, 1880. Each flange was composed of two 2^ x 2^ X i^^ 
steel angles, and one 54- X tf cover plate. The web was a 
12 X -i\"o.5oC" rolled steel plate. The clear span was 5 
feet; pitch of rivet, 4.5 inches; total effective area of section, 
8.51 square inches. The rivet holes were drilled -f-^ inch in 
diameter. 

Referring to Eq. (6) of Art. 65 : 

J. ' jr . th ^ . 

d^ = 12 ms., / = 3.13 sq. ms., -p- = O.375 sq. ms. 

W = centre weight = 130,000 -\- 70 =■ 130,070 pounds. 
M= sWl (/in feet) = 1,951,550. 

Hence: 

^ 1,951,550 ^ J • 

C — '^-^ '-^z — 46,400 pounds. 
42.06 ^ 

The centre load did not break the beam, but caused a 
deflection of 0.9375 inch, and permanent set of 0.50 inch, with 
beginning of side deflection. 

Very closely approximate, / = 252. Hence, with / in feet 
and a centre load of 70,000 pounds with the corresponding de- 
flection of 0.25 inch : 

77 36 X 70,000 X 125 ^ , 

E = '^— — = 5,000,000 pounds. 

0.25 X 252 •^' ^ 

This low value of E is undoubtedly due to the fact that 
the beam was a built one. 



Art. 72.] THICKNESS OF WEB. 60I 

The results of all the tests of built beams given in this 
chapter show that they are much less stiff than rolled ones of 
the same section. In fact, in computing deflections with the 
best designs and best quality of riveted work, E should prob- 
ably never be taken at more than about half its value for 
similar rolled sections, or say at 12,000,000 to 15,000,000. 

After E is determined, the deflection at once results from 
the usual formula : 

^ EI W 

/3 is here in feet, ^is the load at centre, and //the uni- 
form load {L e. weight) of or on the beam. 



Art. 71. — Loss of Metal at Rivet Holes. 

As has been indicated in all examples, the metal punched 
or drilled from parts of beams in tension should always be 
deducted from the total tension area in order to obtain the 
effective area for computation of the ultimate resistance. In 
estimating this loss the actual diameter, as punched or drilled, 
should be taken, and not that of the cold rivet before driving, 
since the latter is always at least one sixteenth inch less than 
the former. 

In the compression portions of the beam, if the work is 
done in a first-class manner, no deduction need be made. 



Art. 72.— Thickness of Web Plate. 

The following approximate method of determining the 
thickness of the web plate in a flanged beam is based upon the 
principles established in Art. 28. 

It was shown in that Article that on two planes which 



602 THICKNESS OF WEB. [Art. ^2, 

make angles of 90° with each other and 45° with the neutral 
surface, and whose intersection forms the neutral axis at the 
section considered, there exists on one a tension and on the 
other a compression, each of whose intensities is equal to that 
of the longitudinal and transverse shear at the same point. It 
was also shown in Art. 17 (see Eq. (38)) that the intensity of 
these shears is f the mean intensity of shear of the whole 
section. 

No essential error is committed (especially in built beams) 
if it be assumed that the whole shear is taken up by the web. 
In the Article just cited it was shown that the intensity of 
shear at the top and bottom surfaces of the beam is zero, as 
well as I the mean at the neutral surface. Now, if this shear 
be assumed uniform in intensity throughout the transverse sec- 
tion of the web, the shear will be made much too large at the 
top and bottom surfaces, and only two-thirds its proper value 
at the centre or neutral axis. 

In accordance with these assumptions on one hand, and the 
established principles on the other, the web may be considered 
as composed of small columns with ends fixed (at the flanges), 
and with sections rectangular, whose axes lie at 45° with the 
neutral surface. 

The assumption of the uniformity of shear in respect to 
these elementary columns causes two errors in opposite direc- 
tions, with the resultant error, in most cases at least, on the 
side of safety. 

In rolled beams, if /' is the mean thickness of a flange, and 
^the total depth, then the length of these elementary columns 
may be taken as : 



or : 



I = {d — 2t') sec 4f, 

I = I.4i4(r/ — 2t') (i) 



In built beams, if d' is the depth from centre to centre of 
rivet holes, there may be taken : 



Art. 72.] THICKNESS OF WEB. 603 

/ — 1.414^' (2) 

If wS is the total shear at any transverse section, A the area 
of that section of the web, taking the depth at d — 2t' or d\ 
and s the mean shear, or : 

5 

then these elementary columns will be subjected to an inten- 
sity of compression equal to s. Hence if /, the thickness of a 
wrought-iron web, is sufficiently great, there may be taken, by 
Gordon's formula : 

s = ^ ....... (3) 

3,000^'' 
Solving this equation for t : 



' - ^ ^ I l,ooo{f - s) (4) 



For the ultimate resistance of wrought-iron rectangular col- 
umns,/ may be taken at 40,000. If a safety factor of 5 be 
taken, the value of / becomes : 



^ = °-°'«3/./s;^^;^^, (5) 



Eq. (5) is for wrought iron only. The empirical constants 
for steel yet remain to be determined. 

These formulae show that / decreases with the depth of the 
beam, and that it also varies in the same direction with s. If, 
therefore, the depth of the beam is constant, Eq. (5) need only 



6o4 THICKNESS OF WEB. [Art. 72. 

be applied at the section where s is the greatest, i. e.y at or near 
the points of support. 

If, however, the depth is variable, it may be necessary to 
apply the formula at a number of sections in order to find the 
greatest value of /. 

Eq. (5) frequently gives much larger values of / than are re- 
quired. It could be made an accurate and valuable formula if 
the empirical quantities which enter it were determined by ex- 
periments on flanged beams. 

The data of Art. 66 give : 

7 
d' = 32 ins., / = -^ in., A = d't = 14 sq. ins. 

5 = 86,650 lbs., .*. J = — = 6,200 (nearly). 

/ = d' X 1.414 = 45-2. 
Hence : 

/ = 1.5 inches (nearly). 

This value with a safety factor of five is evidently excessive, 
though it applies only to the portions RA and HR' of Fig. i in 
Art. 66. Yet, the result may be accepted as indicating that 
the web is certainly too light for those portions, and the ne- 
cessity of the stiffening pieces shown. 

The data of Art. 6y give : 

d' = 27 inches, / = ^ inch, A = d't = 10. 1 sq. ins. 
•S = 26,000 pounds. .'. s = — = 2,600 (nearly). 

/ = d' X 1. 414 = 38.2. 



Art. 72.] GENERAL OBSERVATIONS. 605 



Hence : 



/ = 0.49 inch (nearly). 



The thickness taken, therefore, is probably ample, even 
without the aid of stiffening pieces. . 

The amount of assistance to be derived from stiffeners 
cannot be computed with any certainty. They are very essen- 
tial however, and should be introduced in all large beams. 

However small the built beam, or light its load, the web 
plate should never be less than 0.25 inch in thickness. 

Before leaving this subject it may be well to observe that 
the excessive thickness given by Eq. (5) was, in some measure 
at least, to be anticipated. It has already been stated that the 
assumption of uniform compression throughout the length of 
the elementary column leads to an error on the side of safety. 
Again, the equal tension at right angles to the greatest com- 
pression in the material of the web, as well as the decreasing 
compression toward the centre of the beam, gives support to 
the elementary columns throughout their entire lengths. 
These causes give rise to an excess of safety, in the formula, 
whose amount can only be determined by experiment. Three- 
quarters of the thickness given by the formula would probably 
be ample. 

The experiments of the late Baron von Weber showed that 
a very thin web will give a remarkably large supporting power. 



CHAPTER X. 

Connections. 

Art. 73. — Riveted Joints. 

Although riveted joints possess certain characteristics 
under all circumstances, yet those adapted to boiler and simi- 
lar work differ to some extent from those found in the best 
riveted trusses. The former must be steam and water tight, 
while such considerations do not influence the design of the 
latter, consequently far greater pitch may be found in riveted 
truss work than in boilers. Again, the peculiar requirements 
of bridge and roof work frequently demand a greater overlap 
at joints and different distribution of rivets than would be 
permissible in boilers. 

Kinds of jfoints. 

Some of the principal kinds of joints are shown in Figs. I 
to 6. Fig. I is a ^' lap joint," single riveted ; Fig. 2 is a *' lap 
joint," double riveted ; Fig. 3 is a '* butt joint " with a single 
butt strap and single riveted ; while Figs. 4, 5 and 6 are '' butt 
joints with double butt straps, Fig. 4 being single riveted 
while the others are double riveted. Fig. 5 shows zigzag 
riveting and Fig. 6 chain riveting. All these joints are de- 
signed to resist tension and to convey stress from one single 
thickness of plate to another. Two or three other joints 
peculiar to bridge and roof work will hereafter be shown. 



Art. 73.] 



DISTRIBUTION OF STRESS. 



607 




) 



Fig. 1 







000 



000 



Fig. 2 



Fig. 3 



000 



000 



) 



) o o c 
000 



000 
b o o d 



(f 




c 


) 


c 


D 


([ 


p 










00 






000 
000 









d" 


1) 


c 


b 


c 


) 


q_ 


J) 







Fig.4 



Fig. 5 



Fig. 6 



In the cases of bridges and roofs these " butt straps " are 
usually called " cover plates." 



Distribution of Stress in Riveted Joints. 

A very little consideration of the question will show that 
only an approximate determination of the distribution of stress 
in a riveted joint can be reached. 

In order that rivets, butt straps or cover plates, and differ- 
ent portions of the same main plate may take their proper 
portions of stress, an absolutely accurate adjustment of these 
different parts must be attained ; but all shop work must nee- 



6o8 RIVETED JOINTS. [Art. 73. 

essarily be more or less imperfect, and the requisite condition 
can never be maintained during and after construction. Hence 
the amount of stress carried by each rivet, or each cover plate, 
and hence each portion of the main plate at the joint, cannot 
be found. 

In the cases of lap joints with three or more rows of rivets 
(frequently found in truss work), or in similar work when two 
rows of rivets join a small plate to a much larger one, the out- 
side rows, or row, in consequence of the stretching of the ma- 
terial at the joint, must take far more than their portion of 
stress, if, indeed, they do not carry nearly all. The same con- 
dition of things will exist in butt joints if two or more rows 
are found, under similar circumstances, on the same side of 
the joint. 

If a strip of plate in which the ratio of width over thickness 
is very considerable, be so gripped in a testing machine that 
the applied stress be approximately uniformly distributed over 
its ends, and if it be tested to breaking, it will be found, if the 
broken pieces be joined at the place of breaking, that the cen- 
tral portions of the fracture are widely separated, while the 
edges are in contact. This is due to the cause explained in Art. 
32, " Coefficient of Elasticity." Now if a hole or holes be 
made in or near the centre of the specimen, a portion of the 
material in the front and rear of these holes will be relieved 
from stress, and the total stress in the central section of the 
specimen will be more nearly uniformly distributed in the re- 
maining material. And again, these holes will ** neck " the 
specimen down to a short one. The influences noticed in Art. 
32, " Ultimate Resistance and Elastic Limit,'' will thus be called 
into action. For both these reasons the existence of the hole, 
or holes, in itself, will increase the intensity of the ultimate 
resistance of the plate. 

On the other hand, the effect of the punch, if the hole is 
punched, as will presently be shown, is to decrease the resist- 
ance of the metal about the hole. If the hole is in a joint, also, 



Art. 73.] BENDING IN JOINTS. 609 

the bearing pressure between the rivet and plate is very great, 
and as this pressure must be carried as tension to the material 
adjacent to the rivet hole, and through that in its immediate 
vicinity, the latter {i.e., the material at the extremities of diam- 
ters parallel to the joint) will receive much greater tension than 
that in the central portion between the holes. 

These last two influences tend to reduce the mean intensity 
of ultimate resistance of the material of the joint, and some- 
times more than counterbalance the increase caused by the 
existence of the holes simply as such. In other cases the re- 
sultant effect can only be determined by experiment. 

In Figs. I and 2 it will be observed, that the stresses in the 
plates of a lap joint act excentrically, and, let it first be as- 
sumed, with a lever arm equal to half the sum of the thickness 
of the two plates. If, however, a specimen joint is put in a 
testing machine, the resultant stress may be made to pass 
through the centre of the joint, thus making the lever arm 
for each plate about half its thickness. 

If, therefore, t is the thickness of one plate and /' that of 
the other, while 7" and V are the mean intensities of tension in 
the plates, / the pitch of the rivets and d the diameter ; in the 
first case each plate will be subjected to the bending moment: 

M= Tt{p - d) (l--ti.') = Tf{p - d) (^-^) . (I) 
And in the second : 

M = Te (^-^^\ ; or, T't" (l^) ... (2) 



If K is the greatest intensity of tensile bending stress, then : 
39 



6lO RIVETED JOINTS. [Art. 73. 

The greatest intensity of tension in the plate will therefore 
be : 

T-\-K, ox, T^K (4) 

The moment of inertia / will have the value : 
(/ - d)fi {p - d)f^ 

12 12 

If each plate has the same thickness, t = t' and T = T' \ 
hence : 

ByEq.(i) K=6T {5) 

ByEq.(2) K=^zT (6) 

These values of iTare very large and appear excessive. It 
is to be remembered, however, that the formula used Eq. (3) is 
strictly applicable only within the elastic limit. 

There is no reason to doubt, therefore, that within that 
limit the greatest intensity of tension in the plates of the joint 
may reach from 4 Z to 7 7". 

From these considerations it is to be expected that the true 
elastic limit of the joint, as a whole, would be very low. 

The preceding investigations in the flexure of the joint are 
based upon the virtual assumption that the plates remain 
straight after the application of external stress. In reality 
such a condition of things does not obtain. Even below the 
elastic limit the plates begin to take positions which are 
shown in an exaggerated manner in Fig. 7. On account of the 

bending, the material at the 

points AA stretches much 

-__)P more than that at the points 

BB (with low values of T 
that at the latter points may 
be in compression), so that the centre lines of the plates P and 




Art. 73.] BENDING IN JOINTS. 611 

P' are brought more nearly into coincidence, thus lessening 
the bending moment to which the joint is subjected. After the 
elastic limit of the material at AA is passed, a considerable in- 
crease of strain or stretch takes place at those points for the 
same increment of stress. Two important results follow this 
increase of strain between the elastic limit and failure : the 
joint becomes very markedly distorted, so that the plates /* and 
P' become much more nearly in line, and the stress becomes 
much more nearly uniformly distributed in the sections AB^ 
AB. This is equivalent to saying that the joint is subject to a 
greatly decreased bending moment. 

If the plates are thin, the excess of strain at AA over that 
at BB^ requisite to bring the plates PP' essentially into line, 
may easily be within the stretching capacity of the material. 
If, however, the plates are thick, that condition will not hold, 
and the material at A A will begin to fail before PP' are 
nearly in line. Hence, the mean intensity of stress in a thick 
plate, other things being equal at the instant of rupture, will 
be considerably less than that in a thin one. It might thus 
happen that a lap joint with thin plates would be found 
stronger, even, than one with thicker plates. 

Reference will hereafter be made to experiments w^hich 
verify these conclusions. 

It will now be well to turn back a moment to the consider- 
ation of Eqs. (5) and (6). Those equations show the effect of 
bending to be dependent on T only^ and entirely mdependent of 
the thickness of the plates^ which apparently contradicts the 
conclusion just drawn. But, as has already been intimated, 
those equations involve the virtual assumption that the plates 
remain continually straight, and do not contemplate the altered 
conditions of the joint which exist just at and before rupture. 
Again, they presuppose no passage of the elastic limit. There 
is thus no real contradiction. 

Although a single riveted lap joint only has been treated, 
precisely the same considerations apply to a double riveted lap 



6l2 RIVETED JOINTS. [Art. 73. 

joint, a butt joint with single butt strap or cover plate, and all 
butt straps or cover plates of butt joints. The main plates 
of butt joints with double cover plates are not subjected to 
flexure. 

The rivets of all riveted joints are subjected to heavy flex- 
ure, the greatest of which usually occurs in single lap and butt 
joints like Figs. I and 3. An approximate value of the bend- 
ing moment, in any case, may be found as follows : 

Let n be the number of rows of rivets in one plate. In 
Figs. I, 2, 3, 4, ;^ is I ; and 2 in Figs. 5 and 6. Then if / and f 
are the thickness of the two plates or of one plate and one 
cover, 7" and T' the mean intensities of tension in the same 
pieces, and if M be taken from Eq. (i), the approximate bend- 
ing moment will be: 

M KAd ,^ A . ^ X / X 

— == -^— ; (From Art. 62) ; . . . . (7) 

in which A is the area of the cross section of one rivet, iTthe 
greatest intensity of tension or compression due to bending, and 
d the rivet diameter, as before. From Eqs. (7) and (i) : 

If/=:/': 

^=«^^'^^^ fe) 

This equation is approximate because it Is virtually assumed 
that the pressure on the rivet is uniformly distributed along its 
axis.''^ This is a considerable deviation from the truth, particu- 

* In accordance with this assumption, strictly speaking, \t (thickness of main 
plate) should be taken instead of t in the sum (/ + /') in the above formulae for 
bending, when applied to the double butt joints, Figs. 5 and 6. 



Art. 73.] PRESSURE ON- RIVET. 613 

lariy as failure is approached. The true bending moment is 
much less than that given by Eq. (7) after the rivet has 
deflected a Httle. 

When the joint takes the position shown in Fig. 7, it is clear 
that the rivet is also subject to some direct tension. 

There is a very high intensity of pressure between the shaft 
of the rivet and the wall of the hole. This intensity is not 
uniform over the surface of contact, but has its greatest value 
at, or in the vicinity of, the extremities of that diameter lying 
in the direction of the stress exerted in the plate. At and 
near failure this intensity may be equal to the crushing resist- 
ance of the material over a considerable portion of the surface 
of contact. 

The intricate character of the conditions involved renders it 
quite impossible to determine the law of the distribution of 
this pressure. The bending of the rivets under stress tends to 
a concentration of the pressure near the surface of contact of 
the joined plates, while the unavoidably varying ''fit" of the 
rivet in its hole, even in the best of work, throws the pressure 
towards the front portion of the surface of the rivet shaft. The 
intensity thus varies both along the axis and around the cir- 
cumference of the rivet. 

If any arbitrary law is assumed, the greatest intensity of 
pressure is easily determined. Such laws, however, are mere 
hypotheses and possess no real value. All that can be done is 
to determine, by experiment, the mean safe working intensity 
on the diametral plane of the rivet which is equivalent to a 
fluid pressure of the same intensity against its shaft. 

Thus, if f is this mean (empirically determined) intensity, 
d the diameter of the rivet, and t the thickness of the plate, 
the total pressure carried by one rivet pressing against one 
plate is: 

R^fdf (10) 

There yet remains to be considered the condition of that 



6l4 RIVETED JOINTS. [Art. 73. 

portion of the plate on which the pressure R = fdt is applied, 
and which is situated immediately in front of the rivet. 

This portion of the plate is really in the condition of a beam 
fixed at each end, with a span equal to the diameter of the 
rivet. The beam, however, is not a straight one. At each end 
of the diameter the direct bending stress will be tension ; and, 
on account of the position of the material, its direction will be 
approximately, at least, that of the proper tension of the plate. 
At those points, therefore, the proper and bending tension will 
act to some extent together, and the metal will usually be 
more highly stressed than anywhere else. This accounts for 
the usual manner of tensile fracture of a joint, in which the 
metal begins to tear on each side of the rivets, the metal 
between (generally in a diagonal direction in zigzag riveting) 
being the last to give way. 

In the interior of the joint it is quite impossible to deter- 
mine the value of this tensile bending stress on each side of 
the rivet. On the exterior of the joint, however, an approxi- 
mate result may be reached ; and hence, the depth //, Fig. 2, 
from the centre of the outside row of rivets to the edge of the 

plate. The depth of the beam will be taken as (h j , and 

the pressure or load will be considered concentrated at the 
middle of the diameter or span. If t is thickness of the plate, 
/ the pitch of the rivets and T the mean intensity of tension • 
between the rivets, the load on the beam will be (/ — d) Tt, 
and the moment of inertia of the cross section will be : 

^^ - f) 
i=— — ^. 

12 

From what has been shown in the chapter on bending, 
the modulus of rupture in the present case may be safely 

taken at -T. 
2 



Art. 73.] EFFECT OF PUNCHING. 615 

In Art. 24, the moment at the centre and end of a span 
fixed at each end and loaded in the centre was shown to be 
equal t-o one-eighth the load into the span. 

Hence, by the usual formulae : 

M=i{p- d)Tt =. -^^ =. 3 ^ 'V^ ~ 2 

0. - ^ ^ 



.*. h — o.'ji \^{p — d)d -{- o.^d . . . . (11) 

Reviewing the results of this section, it may be concluded 
that the bending of the plates about axes parallel to them, or 
normal to them in the interior of the joint, and the bending of 
the rivets, as well as the law of the distribution of pressure 
against them, cannot be expressed by formula with any useful 
degree of accuracy ; but that such influences must be recog- 
nized in the empirical determination of the shearing and tear- 
ing resistances of the joint and the mean intensity of pressure 
against the diametral plane of the rivet. 

Effect of Punching, 

The effect of punching wrought-iron plates has been found 
to be injurious. The tensile resistance of the remaining mate- 
rial will be considerably less than that of the plate before 
punching. Yet the injurious effect of the punch does not ex- 
tend far into the plate. If the punched hole is reamed, so that 
the diameter is increased an eighth of an inch, the remaining 
plate will usually give the normal resistance per unit of section, 
or essentially so. 

It has been found by experiment that effect of the punch is 
less injurious as the die hole is increased in diameter, although 
there is probably a limit to the application of this principle. 



6l6 RIVETED JOINTS. [Art. 73. 

The diameter of the die hole is usually from \ to \ larger 
than that of the punch. This excess should depend upon the 
thickness (/) of the plate, and it is sometimes taken as 0\2t. 

Numerous foreign experiments (chiefly English) by Barna- 
by, Stoney, Fletcher, etc., show that the loss of tensile resist- 
ance due to punching wrought-iron plates runs usually from 
10 to 15, and may vary from 5 to 33 per cent, of the original 
resistance. 

The loss of resistance due to puncliing and its remedy, in 
steel plates, have already been treated 'in Art. 34. 

Wrought-Iron Lap Joints^ and Butt Joints with Single Butt 

Strap. 

A butt joint with single butt strap, similar to that shown 
in Fig. 3, is really composed of two lap joints in contact ; since 
each half of the butt strap or cover plate with its underlying 
main plate forms a lap joint. It is unnecessary therefore to 
give it separate treatment. 

From these considerations it is clear that the thickness of 
the butt strap or cover plate should be the same as that of the 
main plate. 

Let t = thickness of plates. 
" d = diameter of rivets. 
'* / — pitch of rivets (i. e., distance between centres 

in the same row). 
** 7" = mean intensity of tension in plates between 

rivets. 
" T = mean intensity of tension in main plates. 
^' / = mean intensity of pressure on diametral plane 

of rivet. 
^' S = mean intensity of shear In rivets. 
^' 71 = number. of rivets in one main plate. 
" ^ = number of rows in one main plate. 
^' h = amount of extreme lap as shown in Fig. 2. 



Art. 73.] LAP JOINTS. 617 

If all the dimensions are in inches, then J", 7^^, /and 5 are 
in pounds per square inch. 

The starting point in the design of a joint is the thickness 
t of the plate. The rivet diameter is then expressed in terms 
of /, and the pitch in terms of the diameter. 

The thickness / of boiler plate depends upon the internal 
pressure, and is to be determined in accordance with the prin- 
ciples laid down in Art. 9, after having made allowance for the 
metal punched out at the holes and the deterioration or other 
effect caused by the punch. 

In truss work the thickness depends upon the amount of 
stress to be carried, and the same allowances are to be made 
for punching and deterioration. 

The relation existing between T and T' is shown by the 
following equations : 

t{p-d)T=tpT :.T^,= ^ ' 



T p -d' 



or. 



7"~ ~7 - ' / 



(12) 



In order that the joint may be equally strong in reference 
to all methods of failure, the following series of equalities must 
hold: 

-tpT'^-t{p-d)T= nfdt = o.ySS4nd'S. 

/. tpT' = t{p - d) T ^ qfdt = 0.7854^^^5. . (13) 

It is probably impossible to cause these equalities to exist 
in any actual joint, but none of the intensities T', 7", /"or S 
should exceed a safe working value. 

In ordinary American boiler practice d varies from 1.5/ to 



6i8 LAP JOINTS. [Art. 73. 

2t\ the latter for thin plates and the former for thicker ones, 
the extreme limits being about | inch and \\ inches. 

The following are some rules given by the best foreign 
authorities for wrought iron : 

Browne <^ = 2t (or 1.25^ with double covers) .... (14) 

Fairbairn </ = 2t for plates less than | in (15) 

Fairbairn d ^^ 1.5/ for plates greater than f in . . . . (16) 

Lemaitre ^ = 1.5/ + 0.16 (17) 

Antoine d = i.i\/t (18) 

Pohlig d = 2t for boiler riveting (19) 

Pohlig d — 3/ for extra strength , (20) 

Redtenbacher .. d — 1.5/ to 2/ (21) 

Unwin d — 0.75^ + A to | / + | (22) 

Unwin d — 1.2V t (23) 

As the results of some of his experiments on ^-inch steel 
plate joints, Prof. A. B. W. Kennedy gives in " Engineering," 
lOth June, 1881, the following rules for rivet diameter : 

Single riveted lap joint .... d = 2.25/ ) 
Double riveted lap joint. . . d = 2.21/ ) 

These rules are for mild steel plates and for greatest 
strength, but are not to be applied to plates over Yz in. thick; 
as the diameters would then become excessive. He therefore 



Art. 73.] 



RIVET DIAMETER. 



619 



THICKNESS 
OF 


DIAMETER OF RIVETS. 






English 












Lloyds' 


Liverpool 




French 


Wilson's 


Hovrez's 


Hall's 


PLATE. 






Dockyard 












Rules, 


Rules. 


Rules. 


Veritas. 


Rules. 


Rules. 


Rules. 


In. 


In. 


In. 


In. 


In. 


In. 


In. 


In. 


-A- 


5 


5. 


1 




5. 


-'4 


h. 


1 6 


8 


8 


•I 


"""" 


8 


lis 


8 


t 


5. 


5. 

8 


ft 

8' 


ft 

8 


\h 


a 
4 


1 


-,^- 


ft 


a 


a 


ft 


a 


13 


_a 


1 b 


8 


4 


± 


8 


4 


1 b 


t 


JL 


a 


11 


a 




y 


1 


ft 


2 


4 


1 (> 


4 


^^ 


4 


8 


16 


9 
1 tj 


3 

4 


it 


i 


a 
4 


i 


I 


I 


5. 


a 
4 


1 


i 




7- 

8 


I 


Ifb 


11 


7 


X 


i 


J. a 


1 






] 


8 


8 


8 


1 ti 


8 




"~~ 


5 


7 


1 ft 




7 








4 


"h" 


K) 




8 







■~^ 


1 3 


7 














T(T 


8 


I 









^~" 





X 


I 


li 


T 1 


I 










w 


I 


1,^0 


T 1 

A 8" 





t1 

■■■ 8 








I 


I 


li 


T 1 

■• 8 


iiV 


T 1 

As" 




~ 



concluded that thicker plates than ^ in. would give propor- 
tionally less resistance. 

It has been found by experiment that there is a very de- 
cided interdependence existing between the values of 7"andy 
in cases of failure by tearing. This is probably due far more to 
the bending action of the rivet, which was considered in detail 
in one of the preceding sections, than to the direct influence of 
the pressure between the rivet and its hole. 

Table I. contains values of 7" and/" at the instant of failure, 
which were tabulated by Prof. Unwin in " Engineering" for Feb. 
20th, 1880. All the plate was English material. The results 
show very clearly the increase of T with the decrease of /". 
They are, however, somewhat discordant. The punched single 
riveted lap joints of Mr. Stoney's experiments show an ap- 
parently abnormally low value of the tenacity 7" for a given 
intensity of compression f\ but the drilled holes show less 
disagreement. 



620 



LAP JOINTS. 



[Art. 73. 



TABLE I. 



Wrought Iron. 



EXPERIMENTER. 



Fairbairn . 



Kirkaldy 



Browne 



Stoney 



FORM OF JOINT. 



Lap, 
Lap, 
Lap, 
Lap, 
Lap, 
Lap, 
Butt, 
Butt, 
Butt, 
Lap, 
Lap, 
Butt, 
Butt, 
Butt, 
Butt, 
Lap, 
Lap, 
Lap, 
Butt, 
Butt, 
Butt, 
f Lap, 
Lap, 
Lap, 
Lap, 
Lap, 
Lap, 
Lap, 
Lap, 
Lap, 
Lap. 
Lap, 
Lap, 
Lap, 
Lap, 



sing^le riveted, 

single riveted 

double riveted 

double riveted 

double riveted 

double riveted 

double riveted 

single riveted, 

single riveted, 

single riveted. , 

double riveted 

double riveted 

double riveted 

double riveted 

double riveted 

ngle riveted . 

ngle riveted. 

ngle riveted. 

ngle riveted 

ngle riveted 

ngle riveted 

ngle riveted, 

ngle riveted, 

ngle riveted, 

ngle riveted, 

ngle riveted, 

ngle riveted, 

ngle riveted, 

ngle riveted, 

ngle riveted, 

ngle riveted, 

ngle riveted, 

ngle riveted, 

ngle riveted, 

ngle riveted. 



, one cover 
two covers 
two covers 



punched . . 

punched., 

punched . , 

punched . , 

punched. 

punched. . 

punched. , 

drilled... 

drilled... 

drilled... 

drilled ... 

drilled... 

drilled..., 

drilled... 



y, IN LBS. PER 
SQUARE INCH. 



83.776 
66,860 
78,290 
76,830 
58,460 

51.300 
58,020 

94, 210 
65,180 
58.580 
36,740 
74,700 
7i>750 
63,170 
62,610 
93,640 
86,950 
84,980 
101,150 
94.240 
92,840 
66,210 
55,660 
49,460 
47,260 
43,680 
42,110 

38,770 
64,400 
59,020 
54,650 
48.370 

47,520 
46,140 
45,920 



Z, IN LBS. PER 
SQUARE INCH, 



39,650 
44,580 
52,190 
48,830 
58,460 
55.330 
53.980 
53,540 
60,700 
47,260 
57,340 
43,900 
45-570 
45,020 
39,200 
29,120 
27,100 
26,300 
31,360 
29,120 
28,880 
31,910 
32,930 
37,630 
35840 
45,920 
44,350 
40,770 
46,820 
34,940 
41,440 
36,740 
47,490 
48.380 
48,270 



Reviewing all the results, it would seem that the following 
values may safely be given single riveted lap joints with 
punched holes in first-class work : 

/ =z 55,000 to 60,000 T = 45,000 to 40,000. 

/ = SSjOOO to 50,000 T = 45,000 to 50,000. 



The following values of/, T and 5, at the instant of failure, 



Art. 73.] PITCH OF RIVETS. 62 1 

are from the experiments (English) of Messrs. Greig and Eyth 
and the Master Mechanics' Association. 

/, IN LBS. PER 7", IN LBS. PER .S", IN LBS PER 

SQ. INCH. SQ. INCH. SQ. INCH. 

r64,400 46,820 40,990"] 

Single riveted 59,490 43,650 41,300 | 

lap joints. . . \ 59,960 43,970 41,680 )■ . .(A) 

62,400 45.760 43,340 

,66,280 47,690 38,77oJ 

All the holes in these joints were drilled, consequently, as 
will hereafter be shown, 5 is a little low. Further, all the joints 
broke by simultaneous shearing of the rivets and tearing of the 
plates : they may therefore be considered well designed. 

Now, ify= T= 50,000, which is experimentally shown to 
be correct in single riveted lap joints, for which ^ = i, the 
second and third members of Eq. (13) give : 

/ = 2d. 

But this pitch would scarcely give sufficient room for heading 
the rivets. It has just been seen that the results in group (A) 
belong to well proportioned joints. An examination of those 
results will show that /" varies from 1.337" to 1.47", nearly; 
which is not an essential disagreement with the results of 
Table I. Hence, putting these values in Eq. (13) : 

/ = 2.33^ to 2.4^ (25) 

This agrees with good ordinary practice in boiler making, 
which makes : 

/ = 2.3^ to 2.y^d, nearly. 

The preceding results are for single riveted lap joints in wrought 
iron. 



622 



LAP JOINTS. 



[Art. 73. 



TABLE II. 

Wrought Iron Double Riveted Lap yoints. 



EXPERIMENTER OR AUTHORITY. 


MODE 
OF RIVETING. 


HOLES 
MADE BY 


POUNDS PER SQUARE INCH 
FOR 




/. 


T. 


Sir Win. Fairbaim 


Hand. 

tt 
t( 
It 
tt 

Machine. 

" 
tt 
It 
tt 
tt 
tt 
tt 

? 

? 


Punch. 

It 
tt 

tt 
tt 
tt 
tt 
tt 

Drill. 

Punch. 

tk 
tt 


68,580 
70,900 
60,860 
69.490 
58,350 
5 ',030 
36,710 
56,380 
53.400 
59,970 
57,030 
34,090 
22,020 
21,540 
21,500 
22,220 
30,330 
3^,230 


51,450 
53,'8o 
45.670 
52,060 
58,350 
54.680 
57,270 
36.470 
34.670 

38,770 
45,790 
49,060 
24,440 
23,630 
28,650 
29,610 
27,280 
28,740 


ii 11 ii 


it (I k( 


11 Ik kt 


ii 11 ki 


(I ii i( 


David Kirkaldy 


Easton and Anderson .. 

i> it tt 

»k 11 It 
Greig and Eyth 




R.V.J. Knight 


It tt 


It tt 


It tt 


It tt 





In the second preceding section considerations were ad- 
duced which show that for a given value of the mean intensity 
of compression between the rivet and its hole, in a double 
riveted lap joint, an increased value of T over that for a single 
riveted lap joint should be expected. So far as comparison 
can be made, Tables I. and II. verify this conclusion, although 

the increase is not very 
great. This arises from 

'J^ the fact that the increased 

length of a double joint 
requires less bending at 
Aj Ay Fig. 8, than a single one to bring the plates P and P' 
nearly into line. 

The tables show that for thin plates / is equal to T, at the 
instant of rupture, for an intensity not far from 55,ocK) pounds 



j:^ (Ou 



n. 



^^ 



A 
TIT" 



Fig. 8 



Art. 73.] PITCH OF RIVETS, 623 

per square inch. This will reduce somewhat the allowable 
ratio between /"and T. 

A careful examination of the results given in the tables 
seems to make it perfectly safe to take/" from i.i7"to 1.257^. 
These values in the second and third members of Eq. (13) give 
(remembering that q is here equal to 2) for double riveted lap 
joints : 

p = 3.2 to 3.5^) 

Or, say : V (26) 

/ = 3.25 to 4.od) 

The smaller values of / belong to thick plates and the 
larger values to thin ones, both because the increased thick- 
ness brings a greater proportional load on the rivet and be- 
cause the lever arm of the bending moment is greater. 

It should be stated that in some apparently good boiler 
practice / is sometimes taken as high even as ^d. The ease 
with which a double riveted lap joint is made steam tight may 
tempt a decrease in expense of riveting. It is probable that 
the rivets of joints in which the pitch exceeds about 4d carry 
an excessive compression and a corresponding liability to 
weakness. 

In Table II. the experiments of Mr. Knight were made on 
plates one inch thick, which are excessively heavy, and the val- 
ues of /"and 7" are remarkably small. It has already been dem- 
onstrated that great thickness of plates would produce results 
of such a character, although the sufficiency of such an expla- 
nation has been doubted. There seems little reason to doubt, 
however, that the cause just cited, together with the normal 
decrease of resistance with an increase of thickness, is a com- 
plete explanation. 

It is to be observed that in the preceding deduced values of 
/ and T, the bending of the plates about axes both parallel 
and normal to their surfaces, have been recognized and pro- 
vided for. 



624 LAP JOINTS, [Art. 73. 

If the accuracy of the experiments cited be assumed, and 
they are the most reliable and valuable that have ever been 
made, there may be taken : 

For inch plates, T — 30,000 to 35,000 lbs. per sq. in. 
For i<(-inch plates, T = 50,000 to 55,000 lbs. per sq. in. 

And for intermediate plates proportional values. 

For single riveted lap joints, y = 1.33 to 1.4 71 
For double riveted lap joints, /"= i.i to 1.25 T. 

As /"and 7" have been found to be dependent on the pecu- 
iar circumstances attending the use of the material in the 
joint, so, in the same general manner, the determination of the 
ultimate shearing resistance of the rivets must involve a similar 
recognition of environment. 

It has been found by experiment, as might have been an- 
ticipated, that rivets in drilled holes offer less resistance to 
shearing than those in punched holes. This arises from the 
fact that the edges of drilled holes are much sharper than those 
formed by a punch. 

Table III. gives the mean results of a large number of ex- 
periments by the authorities named. It has been condensed, 
and the results converted to pounds per square inch, from a 
similar one given by Prof. Unwin, in " Engineering " for 26th 
March, 1880. 

These results are for single riveted lap joints, and therefore 
for single shear. They are only a very little larger than the 
values determined by Chief Engineer Schock for single shear, 
as the apparatus of the latter was essentially equivalent to a 
drilled hole. 

For plates 0.25 inch to 0.375 inch thick, there may be 
taken, as is usually done, 5 = 0.8 7". It has been seen (Table 
II.) that a plate an inch thick can be expected, in lap joints, to 



Art. 73.] 



RIVET SHEARING. 



625 



TABLE III. 

Shearing of Wrought Iron Rivets. 



EXPERIMENTER OR AUTHORITY. 



Fairbairn 

Stoney 

Stoney 

Fairbairn 

Fairbairn 

Master Mechanics' Association 
Greig & Eyth 

Mean result 

Mean result 



KIND OF HOLE. 



Punched. 

Punched. 

Drilled. 

Punched. 

Drilled. 

Drilled. 

Drilled. 



Punched. 
Drilled. 



S IN POUNDS 
PER SQ. IN. 



50,180 
42,200 
40,920 
45,820 
43,610 
46,590 
41,280 



46,030 
43,100 



RESISTANCE (TEN- 

sile) of plate 

OVER >S". 



0.783 
0.910 
I. 061 



I. 071 



0.846 
1.066 



give T not much over 35,000, and as the thickness does not 
seem to appreciably affect S, for this inch plate there may be 
taken S — ^T. The ratio of /"over T has been seen to vary 
from 1.33 to 1.47". Let a mean value of 1.36 for this last ratio 
be inserted in the third member of Eq. (13) ; then, by inserting 
the other values just found in the fourth member of the same 
equation, there will result for single riveted lap joints : 



For thin plates, ^= 2.\t\ 
For thick plates, d = 1.5/ j 



• • (27) 



For double riveted lap joints these results would be dimin- 
ished only slightly. Hence Eq. (27) may be taken as applicable 
to both single and double riveted lap joints in wrought iron. 

It will be observed that Eq. (27) is included within the lim- 
its of the Eqs. (i4)-(23). 

A great number of results by the experimenters already 

cited in this chapter show that the total resistance of a single 
40 



626 LAP JOINTS. [Art. 73. 

riveted lap joint, as a whole, for plates not over 0.5 inch thick, 
may vary from 44 to 58 per cent, of the solid plate in its nor- 
mal condition, and that the mean value may be taken from 50 
to 52 per cent. 

In a double riveted lap joint this mean may be taken at 60 
per cent, of the resistance of the original plate, for moderate 
thicknesses. In Mr. Knight's experiments with inch plates 
(double riveted), the resistance of the joint, as a whole, ranged 
from 33 to 36 per cent, of that of the plate. 

It is clear, from the preceding investigations, that this 
"efficiency "of the joint must decrease as the thickness of 
the plate increases. In fact, Mr. Bertram found, in i860, that 
some joints in ^-inch plates were stronger than those in either 
tV o^ ^-inch plates. Although such results do not involve im- 
possibilities, they are certainly remarkable, and have not since 
been obtained. 

As has before been observed, all the preceding results apply 
directly to butt jointSy in wrought iron, with single butt strap or 
cover plate. 

The width of overlap {li) from the centre of the outside line 
of rivets to the edge of the plate (see Fig. 2) may now be deter- 
mined in terms of d, by the aid of Eq. (ii). Since the load on 
the rivet is represented by (/ — d)Ttj p must be taken in 
terms of d for a single riveted joint, in which/ = 2y^d to 2}^'d. 
As a margin of safety, and as it will, at the same time, simplify 
the resulting expression, let/ = 3</. 

Eq. (11) then gives : 

h = 1.5^. (28) 

Experience has shown that this rule gives ample strength, 
and is about right for caulking, in boiler joints. 

The distance between the rows of riveting is not susceptible 
of accurate expression by formulae, although the considerations 
involved in the establishment of Eq. (11) would lead to an ap- 



Art. 73.] STEEL. 62y 

proximate value. It Is evident, however, that this distance 
should never be as small as h. Apparently, In more than 
double riveted joints, this distance should increase as the centre 
line of the joint Is receded from. In consequence of the bending 
action of the rivet. There are other reasons, however, besides 
that of Inconvenience, why such a practice is not advisable. 

Iji chain riveting the distance between the ce7itre lines of the 
rows of rivets may be taken equal to the pitch in a single riveted 
joints or, as a mean, at 2.^ the diameter of a rivet. 

In sigcag riveting (Fig. 5) this distance may be taken at 
three-quarters its value for chain riveting. 



Steel Lap Joints and Butt Joints with One Cover, 

The general phenomena attending the tests of steel joints 
are precisely the same In kind with those observed in connec- 
tion with riveted iron plates ; they do not, therefore, need par- 
ticular consideration In this section. 

Table IV. contains results communicated to the '* Commit- 
tee of the Institution of Mechanical Engineers " by Messrs. 
Parker and Sharp ('' Engineering," i6th April, 1880). The 
joints failed by tearing, and gave the values of 2" shown In the 
table. The intensity of pressure,/", existed at rupture. 

The following values of T and/ under precisely the same 
circumstances, i.e., failure, were found by Prof. A. B. W. Ken- 
nedy ('' Engineering,** 20th May and loth June, 1881,) for single 
riveted lap joints. 



THICKNESS 

OF PLATE. T. f, 

D-inch 67,060 lbs. per sq. in 42,980 lbs. per sq. in. 

¥ " 65,310" " " " 57,600" " " " 

i " 77,050" " "" 70,850" " " " 

i " 73,030" " "" 70,520" " " " 

f " 80,920" " " " .73,420" " " " 



628 



STEEL LAP JOLNTS. 



[Art. 73. 



TABLE IV. 

Steel Joints. 









HOLE. 


THICKNESS 
OF PLATE. 


POUNDS PER SQ. IN. FOR 




T. 


/■ 


Treble riveted ('rhain'* 


Drilled. 
Punched. 

Drilled. 

Drilled. 
Punched. 


8 in. 
1 in. 
-fb in. 
\ in. 
i in. 
i in. 
i in. 
\ in. 
\ in. 
4 in. 

\ i^- 
4 in. 

I in. 

i in. 

? 

? 


79,220 
52,280 
50,330 
73,360 
70,040 
80,890 
78,400 

66, 940 
67,520 
80,380 
75,780 
63,250 
65,950 
56,760 
87,920 
97,730 


60,010 






< .» 


39-380 
37.740 
57.700 
55,080 
54,470 
52,790 
52,220 
53,330 
73,830 
49.500 
47,580 




i I < 




1 ( ( 




t i ( 




< ( ( 




t t i 




< < ( 




< « < 




< ( ( 




t ( ( 




( (< 


35,460 


Qua 
Doi 
Dov 


idruple 
ible riv 
ible riv 


riveted (zigzag) 

eted butt (one cover) . . . 
eted butt (one cover) . . . 


42,360 
76,200 
83,840 



The holes in these plates were all drilled, and each result is 
a mean of two tests. 

These experiments do not present a sufficient range to 
show clearly the relation existing at failure between T'and/". 
It is clear, however, that no recorded intensity /has been large 
enough to decrease T^to any appreciable amount. In some of 
Prof. Kennedy's tests, in which failure took place by shearing, 
/ was not far from \.2T (with T — 65,000 to 75,000), and it 
would appear from his experiments that such a ratio may prop- 
erly be taken for thin plates in single riveted joints. At the 
same time, with the mild steel used by Prof. Kennedy, T may 
be taken at 70,000 pounds for plates % to }i inch thick. 

Putting 1.2T ior f in the third member of Eq. (13): 



Art. 73.] PITCH OF RIVETS. 629 




Or, say, \ (29) 



for single riveted lap joints. It will probably be best to allow 
this pitch to stand for thick plates also, although experiments 
to verify such a conclusion are yet lacking. For very thick 
plates in single riveting, however, 7" should not be taken over 
50,000 to 55,000 pounds at the highest. 

Experiments on double riveted lap joints by Martell, Kir- 
kaldy and Easton and Anderson, show that it will be essentially 
correct, and certainly safe, to take f and T as in the single 
riveted joints. With q equal to 2, Eq. (13) will then give for 
double riveted steel lap joints : 




Or, say, \ (30) 



Although relating to treble and quadruple riveted joints, 
Table IV. shows in a marked manner the decrease of T with 
the increase of thickness, and verifies the conclusion drawn in 
the preceding section in regard to that phenomenon. 

The results cited by Prof. Unwin, in the report so fre- 
quently referred to heretofore, indicate that for treble riveted 
joints / may be taken essentially equal to T for thin plates, 
and 0.97^ for thick ones. Hence, using Eq. (13) as before: 

TREBLE RIVETING. 

Thin plates (0.25 and 0.375 ii^-)» p = ^d ) 

[ • • (31) 

Thick plates (0.875 ^^<i i-oo iti-)?/ — 3-7^ ) 

Some experiments of Mr. Kirkaldy on joints with J^-inch 
Siemens steel plates quadruple riveted, seem to show that the 
pitch should be about the same as in treble riveted. This is 



630 



STEEL LAP JOINTS. 



[Art. 73. 



undoubtedly due to the fact that with such a great number of 
rivets it becomes impossible to obtain even an approximately 
proper distribution of load among them. 

In treble and quadruple riveting the tests cited show that 
7" may be taken at 70,000 to 75,000 for thin plates, and 55,000 
to 60,000 for thick ones. 

In all the preceding investigations it is supposed that the 
holes are drilled, or that the plates are subsequently annealed 
if punched. 

In nearly all the experiments cited by Prof. Unwin, the 
value of 7", as found in the actual joint, exceeded the ultimate 
resistance of the original plate ; a result which finds its ex- 
planation in the drilling of the holes and the " shortening " 
effect produced by their presence, aided by their equalizing 
effect. 

Table V. gives the ultimate shearing resistance of steel 
rivets as determined by Sharp, Martell, Kirkaldy and Greig 
and Eyth. A very considerable reduction is noticed with the 
increase in plate thickness, due probably to increased bending 
and size of rivet. 

Prof. Kennedy found the following values in single riveted 
lap joints : 



RIVET DIAM. 



o . 75 in 54,460 lbs. per 



1. 00 
1. 00 
0.75 
0.75 
0.75 
0.75 
0.75 



54,400 J 


bs. per sq. in 


37,240 




38,720 




48,030 




49,450 




49,480 




49,300 




47,870 





Each result is a mean of two or three tests. 

In Mr. Kirkaldy's four tests of ^-inch treble and quadruple 
riveted lap joints, with i^-inch rivets, the ultimate shearing 
resistance 5 varied from 41,110 to 46,260 lbs. per sq. in. 



Art. ;3.1 



RIVET SHEARING. 



631 



TABLE V. 
Shearing of Steel Rivets, 



JOINT. 



Single riveted 

Double riveted (chain) 
Treble 



(zigzag). 



Quadruple riveted (zigzag) 



MEAN OF. 



THICKNESS 
OF PLATE. 



4 
I ' 

1 <■ 
8 

X • 
8 

1 • 
8 



iS" IN POUNDS PER 
SQ. IN. 



57.570 

53>6go 
53.310 
50,650 
60,930 
56,220 
57,120 
53,540 
53.980 
43,560 
46,140 
43,010 



Four experiments by Mr, Kirkaldy on single riveted lap joints, during 1881, 
gave .S varying from 52,106 to 54,042 lbs. per sq. in. 

Prof. Kennedy's results give nearly: 

5 — 0.7 7". 
Tables IV. and V., plates not over ^ in. thick : 

5 = 0.8 r. 

Mr. Kirkaldy's for treble and quadruple riveting : 

5=0.77. 

For ordinary plates therefore in single and double riveting, 
for which / = 1.2 7 and 5 as a mean — 0.75 7, the third and 
fourth members of Eq. (13) give : 



d = 2t (nearly) (32) 



632 BUTT JOINTS. [Art. 73. 

For thick plates in treble and quadruple riveting, for which 
/= 0.9 j; and 5 = o.jT\ 

d = i.6t (nearly) (33) 

The rivet pitch, therefore, for steel plates, may be said to 
vary from 2t for thin plates to i,6t for thick ones, with a 
maximum diameter of i\ to 1 1\ inches. 

Prof. Kennedy's best designed single riveted lap joints 
gave from 55 to 64 per cent, the strength of the solid plates. 

Well designed double riveted lap joints should give from 
65 to 75 per cent, the resistance of the solid plate. 

Equally well constructed treble and quadruple riveted 
joints should have an efficiency of 70 to 80 per cent, of the 
solid plate. 

It is therefore seen that there is little economy in more 
than double riveting ordinary joints. 

The distance between the centre lines of the rows of rivets, 
and the distance from the edge of the lap to the outside centre 
line of holes, may be taken the same as for wrought-iron 
joints, according to the rules given in the last part of the pre- 
ceding section. 

All rivets have heretofore been supposed to be steel. In 
the case of steel plates and iron rivets, there may be taken, at 
least approximately, 0.95 for 5, and f = T for thin plates, or 
0.8 7" for very thick ones. These values are to be inserted in 
the preceding formulae for all steel joints, and the results for 
/ and d taken. 

Wrotight-Iron Butt Joints with Double Covers, 

Butt joints with double butt straps or covers differ in two 
respects, and advantageously, from lap joints and butt joints 
with a single cover ; i. e., in the former the rivets are in double 
shear and the main plates are .subjected to no bending. The 



Art. 73.] THICKNESS AND PITCH. 6ll 

cover plates, however, are subjected to greater flexure than the 
plates of a lap joint, for there is no opportunity to decrease 
the leverage by stretching. As the covers form only a small 
portion of the total material, these, with economy, may be 
made sufficiently thick to resist this tendency to failure. 

Let /' = thickness of each cover plate. 

And let the remaining notation be the same as in the pre- 
ceding section. The intensity of compression between the* 
walls of the holes in the cover plates and the rivets, and the 
tension in the former, will be ignored on account of the excess 
in thickness of the two cover plates combined over that of the 
main plate. This excess in thickness is required on account of 
the bending in the covers noticed above. 

TJie thickness of each eover should be from Y^ to y% the thick- 
ness of the main plates, or f = o.y$t to 0.875/. 

The combined thickness of the covers will thus be from 
1.50 to 1.7$ that of the main plates. 

The four principal methods of rupture in the main plate 
will then lead to the following equations, corresponding to 
Eq. (13): 

-tpT = -t{p-d) T= ftfdt = i.syo^nd'S, 

.-. tpT = t {p - d) T = qfdt = i.SyoSgd'S . (34) 

The experiments of Kirkaldy, Fairbairn, Greig and Eyth 
and Knight, show that in well proportioned joints/" = 1.25 to 
1. 5 7" (the higher values belonging to the thinner plates), with 
a mean value of about 1.4 Z". As no bending exists in the 
main plates, this value holds in single or double riveting. 

Hence for single riveting, the second and third members of 
Eq. (34) give 

/ ^ 2.4d; or, say, / = 2.5^ .... (35) 



"34 BUTT JOINTS. [Art. 73. 

In double rivet ingy for which q =- 2 '. 

p = 3.8^; or, say, / = 4.0^ .... (36) 

On account of the essential impossibility of even an ap- 
proximately proper distribution of the load among the rivets, 
and the consequent liability of failure of the joint in detail, in 
^treble riveting the pitch should probably not exceed 4.5^/, nor 
5</ in quadruple riveting. 

There may be taken, according to the experiments just 
cited : 
. For punched inch plates : 

T = 40,000 lbs. per square inch. 
For drilled J<(-inch plates : 

T = 55,000 lbs. per square inch. 

Other thicknesses and conditions give approximately pro- 
portional values, allowing about 10 per cent, for the deteriora- 
tion of the punch ; i.e., T, for a ^ punched plate, may be taken 
at 45,000 pounds. 

It has already been observed that the value of 6" may be 
taken at 0.8 T for lap joints, but the few experiments that have 
been made on shearing in butt joints with double covers, show 
that the ratio must be taken somewhat less, in consequence 
probably of the double shearing which takes place. 

Hence, let 5 be taken at 0.75 T. 

Using the third and fourth members of Eq. (34), therefore, 
and making 5 == 0.75 T : 

i^?r t/iin plates in which f — 1.57": 

^-= 1.3^. (37) 



Art. 73-] STEEL BUTT JOINTS. 635 

For thick plates in which f ^^^ 1.2^ T: * 

^= i.i^. . (38) 

It is hardly worth while, however, to make any rivet less 
than }i inch in diameter. Hence there may be taken the 
limits : 

For i^-inch plate ; d = 0.375 inch. 
For i-inch plate ; d = 1.125 inch. 

These results are verified by good boiler practice. 

The distance from the centre line of outside row of rivets 
to the edge of the cover plate, or from the edge of the main 
plate to the centre line of the first row of rivets in the same, 
may be taken at ^d as in lap joints, since the calculation is 
precisely the same. This rule frequently gives a considerable 
margin of safety over that of any other portion of the joint. 

The distance between the centre lines of the rows of rivets 
may be taken at 2.5 to ^.od for chain riveting, and ^ that dis- 
tance for zigzag riveting. 



Steel Butt Joints with Double Cover Plates, 

For the same reasons stated in the preceding section, con- 
siderations touching the stress in the cover plates will be 
omitted. And also, for the reasons there given, these cover 
plates should each possess from J4^ to ^ the thickness of the 
main plate ; or : 

/' = 0.75 to 0.875/. 

Table VI. gives the results of a large number of tests in 
which the joint failed by the tearing of the plates. The in- 
tensities of tension and compression, 7" and/*, existed at failure. 



636 



STEEL BUTT JOINTS. 



[Art. 73. 



TABLE VI. 
Double Riveted Butt Joints. 



KXPERIMENTER OR AUTHORITY. 



Henry Sharp . 
Martell ! 



Boyd 

Kirkaldy, annealed plates . 



Greig and Eyth . 
Parker 



Kirkaldy, 8 inch Siemens steel plates. Mean of two, 

" lii rivets 



Landore 





POUNDS PER 


SQUARE INCH 




FOR 


HOLES BY 








T. 


/. 


Drill. 


96,160 


83,330 


Punch. 


87,600 


75,170 


Drill. 


55,100 


76,205 


'* 


5I1740 


71,680 




64,290 


88,890 




58,690 


89,130 




55.200 


76,160 




51,230 


70,800 




64,320 


88,930 


Punch. 


68,990 


93,t6o 


'' 


75,490 


101,900 




82,450 


99,660 




83,180 


100,510 




76,590 


90,460 




78,220 


92,380 




74,030 


92,850 




70,540 


88,500 




73,920 


84,630 




72,560 


83,080 


" 


72,390 


107,110 




76,520 


112,780 


Drill. 


67,670 


92,270 




57,360 


71,440 


" 


49'370 ■ 


49,100 




50,920 


50,760 




62,140 


107,410 


(( 


61,800 


74,520 




66,200 


112,000 




63,260 


107,700 


Bored. 


63,560 


104,100 


" 


69,590 


117,300 


Punch. 


67,540 


98,630 


" 


66,750 


108,000 


Bored. 


67,260 


121,060 



The first of the last set of results in the table, by Mr. Kirkaldy, 
was found with zigzag riveting in which the distance between 
the centre lines of the rows of rivets was too small. 

These results are quite irregular, but it would seem to be as 
safe a deduction as possible to take/— 1.25 7", with T equal to 
70,000 to 75,000 pounds per square inch for thin plates, and 
55,000 to 60,000 for thick ones. 



Art. 73.]^ PITCH AND RIVET SHEARING. 637 

With this value of/, and q — 2, the second and third mem- 
bers of Eq. (34) give for double riveted butt joints with two 
covers : 

P = 3-5^. (39) 

If the same value of / be preserved, there will result for 
single riveted butt joints with two covers : 

p — 2.^d. (40) 

Experiments on treble and quadruple riveting are yet lack- 
ing. 

But few experiments on the shearing of rivets in butt 
joints with double covers have yet been made. Four tests by 
Messrs. Sharp and Kirkaldy give : 

, THICKNESS 

OF PLATE. 5". 

Single riveted 42,000 lbs. per sq. in. 

Double " 0.875 in 44.550 " 

•' 53,870 •' 



0.55 in 42,700 " 

0.875 in 44,420 *' 



( ( ( < (< 
<« i < ( ( 
t < ( ( ( ( 



All the holes were drilled. 
* These values of 5 range about 0.77". Putting this ratio, 
therefore, in Eq. (34), and taking / ::= 1.25 7", the third and 
fourth members of that equation give : 

d = 1.14/ (41) 

It is probable that this is a little too small for thin plates, 
and a little too large for thick ones. Hence there may be 
taken : 

For thin plates, d = i]^t ] 

\ . . . . (42) 

For thick plates, d = lyit ) 



638 



EFFICIENCIES OF JOINTS. 



[Art. 73. 



Double riveted butt joints designed in accordance with the 
foregoing deductions should give a resistance ranging from 65 
to 75 per cent, of that of the solid plate. 

Single riveted joints will give an efficiency somewhat less ; 
perhaps from 60 to 65 per cent. 

It is to be supposed, in applying the rules just established, 
that all steel plates are drilled, or subsequently annealed if 
punched. 

As in the preceding cases, the distance between the centre 
lines of the rows of rivets may be taken at 2.5 to 3</ for chain 
riveting, and three-quarters that distance for zigzag. 



Efficiencies, 

The values of the quantity which has been termed the 
"efficiency" of the joint, i.e.^ the ratio of the resistance of a 
given width of joint over that of an equal width of solid plate, 
in the preceding investigations, are those actually determined 
by experiments with the joints themselves. They may, there- 
fore, be relied upon. Some values which have for many years 
been considered as standard, but which, in reality, are of a 



TABLE VII. 
Butt Joints with Two Covers — 1877. 



NO. OF 


PLATE 


RIVET DI- 


PITCH OF 


HOLES. 


RIVETING. 


EFFICIENCY. 


TESTS. 


THICKNESS. 


AMETER. 


RIVETS. 








2 


-h in. 


t in. 


2^ in. 


Punched. 


Chain. 


0.672 


2 


-h in. 


«- in. 


3 in. 


Punched. 


Zigzag. 


0.669 


2 


\ in. 


fin. 


24 in. 


Drilled. 


Chain. 


0.662 


2 


\ in. 


1 in. 


3 in. 


Drilled. 


Zigzag. 


0.633 



Art. 73.] 



EFFICIENCIES OF JOINTS. 



639 



somewhat arbitrary nature, and at best belonging to a limited 
class of joints, have been disregarded. 

Table VII. gives the results of Mr. Kirkaldy's experiments 
in reference to the comparative resistance of chain and zigzag 
riveting. The difference is not great, but what there is is in 
favor of the chain riveting. 

TABLE VIII. 



Kirkaldfs Tests — 1872. 



JOINT. 



Lap 

Lap 

Lap 

Lap 

Butt, I cover . 
Butt, I cover . 
Butt, I cover . 
Butt, I cover . 
Butt, 2 covers, 
Butt, 2 covers. 
Butt, 2 covers, 
Butt, 2 covers. 



Single. 

Single. 

Double. 

Double. 

Single. 

Single. 

Double. 

Double. 

Single. 

Single. 

Double. 

Double. 



Punched. 

Drilled. 

Punched, 

Drilled. 

Punched. 

Drilled. 

Punched. 

Drilled. 

Punched. 

Drilled. 

Punched. 

Drilled. 



RIVET DIAMETER 


IN TERMS OF t. 


d = 


2t 


d = 


2/ 


d = 


It 


d ^ 


2/ 


d^ 


2t 


d^ 


2t 


d = 


It 


d ^ 


It 


d^ 


^\t 


d^ 


^\t 


d - 


11/ 


d = 


li/ 



PITCH IN 
TERMS OF d. 



p = 3d 
p^Q^d 
p = ^\d 
p = ^d 
P = 2>d 
p = 2%d 
p=^U 
p=/^d 
P = 'i\d 
p-3d 
P = 5{d 
p = 4ld 



EFFICIENCY. 



0.55 
0.62 
0.69 
0.75 
0.55 
0.62 
0.69 
0.75 
0.57 
0.67 
0.72 
0.79 



Table VIII. gives the results of the same experimenter on 
the relative value of punched and drilled work. 

The drilled work is seen to give decidedly the greatest 
efficiency in every case. 

The joints to which Tables VII. and VIII. belong were of 
wrought iron. 

Experiments by Mr. Kirkaldy during 1881 show that well- 
designed double riveted steel butt joints with two covers may 
be expected to give efficiencies varying from 0.65 to 0.75. 



640 



TRUSS JOINTS. 



[Art. ^z. 



Riveted Truss Joints. 

The circumstances in which riveted joints are used in truss 
work, render permissible many special forms which can find no 
place in boiler riveting. If joints are found under the same 
circumstances, so far as the transference of stress is concerned, 
precisely the same forms would be used, except that caulking 
is, of course, only required in boiler work. 

Fig. 9 shows a common form of chord construction in riv- 



^ r\ r\ r\ r\ r-\ n\ r\ r\ r\ r\ ^\ r\^ 



jO^ 



.C^ 



o^Yri^tr^ 



w ^^w-^^v^w 




.oZn^ 



0^0010 o;:o 




*^ Fig. 9 

eted truss work, with the relative proportions 
exaggerated. 

The lower portion of the figure shows a 
section of the chord, in which the cover 
plates are shaded. The joint is supposed to 
be in tension. 

AB is a horizontal cover plate, under 
which the horizontal component plates form 
lap joints at C, D and E. As the distance 
MN must necessarily be much greater than the allowable pitch 
in boiler work, these lap joints, considered in themselves, 
should be at least treble riveted. On the other hand, the pre- 
ceding investigations show that even with treble riveting there 
is great disparity in the loads carried by the different rivets 
and consequent tendency to detailed rupture; there would 




Art. 73-] TRUSS JOINTS. 641 

seem, therefore, to be little or no benefit in more than treble 
riveting. 

The distance between the centres of rivets along the line of 
the chord — i.e.y along AB in the upper figure — may be taken at 
three diameters. The overlap CD = DE (upper Fig.) would 
then be taken at 9 diameters, and from A, C, D ox E to the 
centre of the first hole, at i^ diameters. The cover AB 
should extend 9 diameters also on either side of C and E. 

In this work the diameter of the rivet may usually be taken 
about the same as for boiler work. In estimating the resist- 
ance of the whole joint, however, it is to be borne in^ mind 
that the rivet holes take metal out of all the plates, and that 
they are usually punched. 

It is impossible to follow the stresses in such a joint or to 
compute its eflficiency. If tested to failure, the latter would 
probably be found pretty low. 

The joint in the vertical plate should be formed as at FG — 
i.c.y it should be a double cover butt joint. The principles al- 
ready established in a preceding section, in regard to the thick- 
ness of covers and diameter of rivets, should be observed here. 

The two rows of rivets on either side of the joint may as 
well be chain riveted with a pitch 3^ to 4 diameters. Other 
rivets should then be staggered in until the group of rivet cen- 
tres on each side is brought to a point, as shown in the up- 
part of Fig. 9. In this manner the available section of a width 
of plate equal to that of the cover, becomes approximately 
equal to the total, less the material from one rivet hole. Hence 
the efficiency of the joint becomes correspondingly increased. 

If the joint is in compression the preceding observations 
hold without change, except that all covers should have the 
same thickness as the plates covered. 

Even if the joints C, D, E and H are of planed edges, little or 

no reliance should be placed upon their bearing on each other, 

since the operation of riveting will draw them apart more or 

less, however well the work may be done. Melted zinc, or 

41 



642 FRICTION OF RIVETED JOINTS. [Art. 73. 

other similar metal, has been poured into compression joints 
with the intention of insuring good bearings, but the results 
are not satisfactory. 

In the case of very wide chords, four longitudinal rows of 
rivets should be used in such joints as are exemplified in 
Fig. 9. 

Unless great caution is observed and excellence of design 
secured, there will frequently be excessive bending in the 
riveted joints of trusswork, on account of the great variety of 
connections required. 

Diagonal Joints, 

It has been proposed to form riveted joints, the edges of 
whose plates are neither perpendicular nor parallel to the 
stress transferred. In this manner a greater number of rivets 
and a greater section of metal will resist the stress exerted in 
the body of the plate. 

Mr. Kirkaldy made some tests on such lap joints, single 
riveted, with ^-inch plates, the joints of which lay at 45° with 
the applied force, with the following results : 

Entire plate 100 

Square joint 59-4 

Diagonal joint 87.2 

The diagonal joints are thus seen to give by far the best 
results. They are, however, much the most expensive also. 



Friction of Riveted Joints, 

There are not lacking experiments to show that the friction 
between the plates of a riveted joint is very great. This, how- 
ever, cannot be relied upon to give additional resistance to the 
joint, since a sensible relative movement of the plates takes 



Art. 74,] WELDED JOINTS. 643 

place in advance of its greatest resistance and essentially de- 
stroys the friction. 

The experiments of Edwin Clarke, Harkort and Lavelley 
show that this friction may range from 8,330 to 22,400 lbs. 
per sq. in. of rivet section. 

The specimens were prepared with one slotted plate, so 
that friction was the only resistance to the parting of the 
plates. 

Hand and Machine Riveting, 

Pneumatic, steam and hydraulic riveting machines have 
lately been brought to such a degree of perfection, that ma- 
chine work is now very generally preferred to hand riveting. 

The resistances of joints will vary to some extent with the 
method of riveting. Usually, however, the variation will not 
be greater than may be found for the same kind of riveting in 
different places and under different circumstances. 

As a rule, machine riveting is much more reliable than 
hand, in that, the hole is better filled and the rivet more 
quickly headed, in consequence of the great excess of pressure 
exerted. There is thus much less liability of loose rivets. 

Many of the preceding experimental results were obtained 
from machine work. 



Art. 74. — Welded Joints. 

At the present time the process of welding can, with proper 
care and material, be made to give excellent results. 

Scarf welds give much better results than lap welds, on 
account of the bending to which the latter are subjected. 

Mr. Kirtley (Institute of Mechanical Engineers of Great 
Britain) made some experiments with small strips, 7.5 inches 
long and jV inch thick, cut across welded joints. These strips 



644 PIN CONNECTION. Art. 75.] 

were taken out of boilers whose longitudinal joints had been 
welded. Twenty-three experiments with strips varying from 
one to one and a half inches wide, gave the following results 
per square inch of plate section : 

WELDED. SOLID PLATE. 

Greatest 53,310 lbs 57,790 lbs. 

Mean 46,140 " 52,860 " 

Least.... 36,960 " ... 46,370 " 

The mean result is seen to be nearly 90 per cent, of that 
of 'the solid plate. 

Although the test specimens were altogether too small to 
be of the greatest value, the results are most excellent. 

The value of a welded joint depends as well upon the 
nature of the material welded as upon the manipulation during 
welding. 

Art. 75. — Pin Connection. 

A pin connection consists of two sets of eye bars or links, 
through the heads at one end of each of which a single pin 
passes. Fig. i shows a pin connection; A, A^ B, B, 3.re eye 
bars or links, and P is the pin. 




Fig.l 



The head of the eye bar (one is shown in elevation in Fig. 
2) requires the greatest care in its formation. It is imperfect 



Art. 75.] 



EYE BAR HEADS. 



645 



unless it be so proportioned that when the eye bar is tested to 
failure, fracture will be as likely to take place in the body of 
the bar as in the head — in other words, unless its efficiency is 
unity. 

In Fig. 2 the head of the eye bar, or link, is supposed to be 
of the same thickness as that of the body of the bar whose 
width is w. 



^ 




B-4-G- 



M K 



If t is the thickness of the bar, so that wt is the area of its 
normal section, then t is almost invariably included between 
the limits of y^w and yiw. In fact these extreme values are 
each too extreme for the full resistance of the bar, although 
they are sometimes used. These ratios, as well as the diameter 
of the pin in terms of w^ can only be determined by experi- 
ments on full-sized bars. A large number of such experiments 
have been made both in this country and in Great Britain, and 
while the resistance of the bar as a whole depends to a con- 
siderable extent on the mode of manufacture or formation of 
the head, it has been found that for the best proportioned head 
t should range from y^w to Y^w, and the diameter, <-/, of the 
pin from o.J^iv to w. 

It is extremely difficult to reach more than a general idea 
of the condition of stress in an eye bar head, although an ap- 
proximate mathematical treatment of the question may be 
found in the ''Trans. Am. Soc. of Civ. Engrs.," Vol. VI., 1877, 



646 PIN CONNECTION. [Art. 75. 

in which the results agree essentially with those of experi- 
ment. 

Before taking a general view of the stresses which may- 
arise in an eye bar head, it must be premised that a difference 
of -g^' to 1 Jo-" between the diameter of the pin and that of the 
pin-hole is exceptionally good practice. Before the eye bar is 
strained, therefore, there is a line of contact only between the 
pin and eye bar head, but on account of the elasticity of the 
material, this line changes to a surface when the bar is under 
stress, and increases with the degree of stress to which the bar 
is subjected. The line and surface of contact is, of course, in 
the vicinity of Q^ Fig. 2, i.e.^ on that side of the pin toward 
the nearest end of the bar. The consequence of this is, that 
when the bar is strained, the portion about QB, Fig. 2, is sub- 
ject to direct compression and extension ; that about RL, NE 
and GS to direct tension and bending, while in the vicinity of 
T there is a point of contra-flexure, and the stress in the direc- 
tion of the circumference changes from compression to tension 
as E is approached from Q. 

As a result of many of the experiments which have been 
made, the following mode of proportioning the head has here- 
tofore been very extensively used : Let r represent the radius 
of the pin, while reference is made to Fig. 2. Then take 
EN = o.66w. The curve DRBK is a semicircle with a radius 
equal to r -|- o.66w^ with a centre, A^ so taken on the centre 
line of the bar that QB = o.^'jw. GF is a portion of the same 
curve, with ^' as the centre {A'C = AC); GUIs any curve 
with a long radius joining 6^7^ gradually with the body of the 
bar. HG should be very gradual in order that there may be a 
large amount of metal in the vicinity of CG, for there the 
metal is subjected to flexure as well as direct tension. FD 
is a straight line parallel to the centre line of the bar. 

As the preceding rule gives a head whose outline causes a 
more expensive die than a simple circle, at the present day 
eye bar heads are usually formed as shown in Fig. 3. 



Art. 75.] 



EYE BAR ITEADF;. 



647 




ABD is a semicircle with a radius equal to ;* -f o.Zw to 
r + o.Qee^, and whose centre C is the centre of the pin-hole. 
The portions FA and HD are formed as before. 

There should be no weld across the bar in the vicinity 
of FH. Consequently, heads are usually formed by placing 
proper sized pieces upon the upset ends of the plain bars, and 
then, after insertion in a heating furnace, forcing the head to 
the desired shape in a die under hydraulic or steam pressure. 

The intensity of this pressure will affect, to a considerable 
extent, the permissible dimensions of the head. The greater 
the pressure, the better will be the results. 

The unfinished head is sometimes rolled on the bar, as by 
the Kloman process. 

The thickness of the head is sometimes made greater than 
that of the body of the bar. If the head is circular, as in Fig. 
3, the section of metal on each side of the pin (through AC ox 
CD) should be not far from eight-tenths that in the body of 
the bar. 

This thickening of the eye bar head is an excellent thing 
for the bar, but subjects the pin to a great increase of bending, 
and in that respect very injurious. 

In pin connections, the pin is subjected to very heavy 
bending.'^ 



* For a detailed treatment of this subject, the author's "Bridge and Roof 
Trusses " may be consulted. 



648 CABLES OR ROPES. [Art. "J^. 

If M is the bending moment to which the pin is subjected, 
K the greatest intensity of bending stress developed, and A 
the area of the normal section of the pin, Eq. (4) of Art. 63 
gives : 



A/J 
J/ =z /^lir = 0.1 AV3 (nearly) . . . . (i) 




Or: 



JM 
d= 2.16 y;^ (2) 

Values of iT, for circular sections, may be found in Art. 63. 



Art. 76. — Iron, Steel and Hemp Cables or Ropes. — Wrought-Iron Chain 

Cables. 

The following tables of resistance and other properties of 
cables are those published by John A. Roebling's Sons Co. 

It will be observed that the figures for hemp ropes are 
given in comparison with either iron or steel in each of the 
tables. 

In considering the resistance of iron and steel cables com- 
posed of wire twisted into strands, it is of the highest impor- 
tance to keep clearly in view the circumstances or conditions 
produced by the manner of fabrication, as they are peculiar to 
all classes of ropes, whether of hemp or wire. 

In this class of material the fibres or strands no longer lie 
parallel to the direction of the stress which they carry, but the 
process of twisting causes each fibre or wire to take a helical 
form, the pitch of which is not constant for the different por- 
tions of the rope. The consequence is that if the process of 
fabrication were absolutely perfect, so that each wire or fibre 
could take its proper portion of load, the stress in that wire or 
fibre would be its portion of load multiplied by the secant of 



Art. 'je:] 



CABLES OR ROPES, 



649 



its inclination to the axis of the rope. As a matter of fact, 
however, each wire does not take its proper portion of load ; 
the imperfections unavoidably incident to the processes of 
manufacture render such a result impossible. Hence the in- 
creased necessity of experimental determination of the ulti- 
mate resistances of metallic and hemp ropes. 

The same composite character of these productions renders 
anything like an approximately elastic character, even, an 
essential impossibility. It is true that any rope will yield to a 
considerable extent while under stress, and then return nearly 
to its original condition, but this behavior is only apparently 
elastic ; it is almost entirely due to the increase of helical pitch 
of the strands caused by the external loading. During this 

Standard Hoisting Ropes with 19 Wires to the Strand. 





c 




^ C 


B 8 


king 
IS of 
ds. 


JS 








^ 2 ^ 


rt 


V to 
a 




c 




h •« & 


en v„ 


c 


s s- s 


Trade No. 


1 1 

3 


Diameter, 


^ en J5 





!^ - 

w "o 8 
a rt 

S -2 fT 


ii-g 




U 




M 


Ph 


U 


I 


61 


24 


8.00 


74 


15 


15^ 


2 


6 


2 


6.30 


65 


13 


14I 


3 


5i 


If 


5-25 


54 


II 


13 


4 


5 


Ig 


4. 10 


44 


9 


12 


5 


4^ 


li 


3-65 


39 


8 


Hi 


5i 


4l 


l| 


3.00 


33 


6* 


loi 


6 


4 


4 


2.50 


27 


5i 


9i 


7 


3^ ■ 


li 


2.00 


20 


4 


8 


8 


3i 


I 


1.5S 


16 


3 


7 


9 


2I 


8 


1.20 


Hi 


2i 


6 


10 


2l- 


f 


0.88 


8.64 


II 


5 


lOi 


2 


8 


0.70 


5-13 


li 


4i 


\o\ 


I| 


A 


0.44 


4.27 


f 


4 


io| 


li 


i 


0.35 


3.48 


i 


3i 



650 



CABLES OR ROPES. 



[Art. J6, 



Standard Hoisting Ropes with 19 Wires to the Strand. 



CAST STEEL. 









foot 
rope 
cen. 


■S 8 

"rS 


king 
IS of 
ds. 


nee of 
pe of 
sngth. 




c ,f. 




u ^ 0, 


<fl ,„ 


G 


Trade No. 


% 1 

el 



Diameter. 


^ S 

:£ a ^ 

.S ^ 




III 

§.sa 


te ^ = 

!•- ^ 

nj "o 8 
a. rt 
2 ° cT 


3 C 3 
>3 X! <U 




u 




CQ 


PU 


U 


I 


61 


2l 


8.00 


130 


26 






2 

3 


6 

5i 


2 


6.30 
5-25 


100 

78 


21 
17 




I5i 


4 


5 


If 


4.10 


64 


13 


14^ 


5 


4? 


li 


3-65 


55 


II 


13^ 


6 


4 


li 


2.50 


39 


8 


Hi 


7 


3i 


13 


2.00 


30 


6 


10 


8 


3i 


I 


1.58 


24 


5 


9i 


9 


24 


8 


1.20 


20 


4 


8 


10 


2i 


4 


0.88 


13 


3 


6i 


loi 


2 


8 


0. 70 


9 


2 


5i 


104- 


15 


-r6 


0.44 


63L 


li 


4l 


io| 


li 


\ 


0.35 


5i 


I 


4i 



operation the strands endeavor to place themselves more 
nearly parallel to the direction of stress, and give rise to a cor- 
responding decrease in diameter. Since these influences pre- 
clude the existence of either coefficient of elasticity or elastic 
limit, ultimate resistances only will be given in this section. 

The preceding observations evidently do not apply to sus- 
pension bridge cables which are built up of parallel wires. The 
operations leading to the production of such a cable are of 
such a refined and exact character that the total resistance of 
the cable may be assumed without essential error to be the 
sum of the resistances of all the wires taken separately : the 
coefficient of elasticity and elastic limit may, and usually do 
exist with perfect definition. 



Art. j6.'\ 



CABLES OR ROPES. 



651 



Galvanized Steel Cables for Suspension Bridges. 





ULTIMATE STRENGTH IN TONS OF 




DIAMETER IN INCHES. 


2,000 POUNDS. 


WEIGHT PER FOOT. 


2| 


220 


13 


2i 


200 


II-3 


2f 


180 


10 


2l 


155 


8.64 


2 


IIO 


• 6.5 


ll 


100 


5.8 


If 


95 


5.6 


If 


75 


4.35 


li 


65 


3-7 



Transmissioji and Stattdifig Ropes with 7 Wires to the Strajtd. 



Trade No. 


.s 



c 

^ Si 


Diameter. 


per foot 
s. of rope 
hemp cen. 


Breaking strain 
in tons of 2,000 
pounds. 


working 
in tons of 
pounds. 


ference of 

rope of 

strength. 




6 S 

3 




Weight 
in lb 
with 


Proper 
load 
2,000 


Circum 
hemp 
equa] 


II 


4^ 


n 


3-37 


36 


9 


IO| 


12 


4i 


li 


2.77 


30 


7i 


10 


13 


3l 


li 


2. 23 


25 


6i 


9i 


14 


3» 


li 


1.82 


20 


5 


8 


15 


3 


I 


1.50 


16 


4 


7 


16 


2^ 


i 


1. 12 


12.3 


3 


6i 


17 


2i 


4 


0.88 


8.8 


2i 


5i 


18 


28 


H 


0.70 


7.6 


2 


5 


19 


l| 


5. 

8 


0.57 


5.8 


l\ 


4l 


20 


If 


"lb 


0.41 


4.1 


I 


4, 


21 


If 


i 


0.31 


2.83 


a 
4 


3i 


22 


li 


ftr • 


0.23 


2. 13 


i 


2| 


23 


li 


f 


0.19 


1.65 


— 


2^ 
24 


24 


I 


-5.. 
1.^ 


0.16 


1.38 


— 


25 


1 


fi 


0.125 


1.03 




2 



652 



CHAIN CABLES. 



[Art. 'je. 



Transmissio?t and Standing Ropes with 7 Wires to the Strand. 



CAST STEEL. 





CJ 




4J 1) . 


c 


hr «<-' 


H-( «tH • 









Q, C 


rt 8 


c . 


J3 

*J .. be 
u e 




s ,«• 




w ^ 0, 


</> ,^ 


fl 


Trade No. 


3 
u 


Diameter. 


Weight pe 
in lbs. 
with hem 


rcaking 
in tons 
pounds. 


roper w< 
load in t 
2,000 pou 


ircumfere 
hemp re 
equal str 




U 




CQ 


Ph 


U 


11 


4f 


't 


3.37 


67 


16 


15 


12 


4l- 


l| 


2.77 


55 


12^ 


13 


13 


31 


li 


2.28 


45 


10 


12 


14 


38^ 


li 


1.82 


36 


8 


io| 


15 


3 


I 


1.50 


30 


6.} 


10 


16 


2^ 


1 


I. 12 


22 


5 


8i 


17 


2| 


I 


0.8S 


17 


3h 


7i 


18 


2:^ 


1 D 


0.70 


I3i 


3 


6^ 


19 


ll 


1 


0.57 


10 


2I 


5^ 


20 


Is 


T"6 


0.41 


8 


U 


5 


23 


if 


i 


0.31 


6 


li 


4l 


21 


li 


8 


0. ig 


4 


I 


3t 


24 


I 


A 


0.16 


3 


3 

4 


3k 



Wrought-Iron Chain Cables, 

It might at first sight be supposed that the pull which the 
link of a chain cable could resist would be twice that offered 
by a bar of round iron equal in cross section to that of one side 
of the link. But a weld exists at one end of the link and a 
bend at the other, each requiring at least one heat for the por- 
tion of the link in which it is located. These manipulations 
produce a considerable decrease in the resistance of the link. 

The United States Committee on " Tests of Chain Cables," 
of which Commander L. A. Beardsley was chairman, made 
many experiments on the iron of which chain cables are made, 
as well as on the finished cables. 



Art. 'J6^^ 



CHAIN CABLES. 



653 



The following conclusions and table are taken from the 
report of that committee : ^* . . . that beyond doubt, when 
made of American bar iron, with cast-iron studs, the studded 
link is inferior in strength to the unstudded one. 

Ultii7iaie Resistance and Proof Tests of Chain Cables. 





AV. KESIST. = 


PROOF 




AV. RESIST. = 


PROOF 


DIAM. OF BAR. 






DIAM. OF BAR. 








162,% OF BAR. 


TEST. 




163;? OF BAR. 


TEST. 


Inches. 


Pounds. 


Pounds. 


Inches. 


Pounds. 


Pounds, 


I 


71,172 


33,840 


i-rb 


162,283 


77,159 


^-h 


79.544 


37,820 


I^^ 


174,475 


82,956 


li 


83,445 


42,053 


•lU 


187,075 


88,947 


IT^ 


97,731 


46,468 


i! 


200,074 


95,128 


li 


107,440 


51,084 > 


1^1 


213,475 


101,499 


lA' 


117,577 


55,903 


ll 


227,271 


108,058 


If 


128,129 


60,920 


T-'-A- 


241,463 


114,806 


I-.v 


139,103 


66,138 


2 


256,040 


121,737 


li 


150,485 


71,550 









" That, when proper care is exercised in the selection of 
material, a variation of five to seventeen per cent, of the 
strongest may be expected in the resistance of cables. With- 
out this care, the variation may rise to twenty-five per cent. 

" That with proper material and construction the ultimate 
resistance of the chain may be expected to vary from 155 to 
170 per cent, of that of the bar used in making the links, 
and show an average of about 163 per cent. 

" That the proof test of a chain cable should be about 50 per 
cent, of the ultimate resistance of the weakest link." 

The decrease of the resistance of the studded below the 
unstudded cable is probably due to the fact that in the former 
the sides of the link do not remain parallel to each other up to 
failure, as they do in the latter. The result is an increase of 
stress in the studded link over the unstudded in the proportion 



^54 CHAIN CABLES. [Art. 'J^. 

of unity to the secant of half the inclination of the sides of the 
former to each other. 

From a great number of tests of bars and finished cables, 
the committee considered that the average ultimate resistance, 
and proof tests of chain cables made of the bars, whose diam- 
eters are given, should be such as are shown in the accompany- 
ing table. 



CHAPTER XL 

Miscellaneous Problems. 

Art. 77. — Resistance of Flues to Collapse. 

If a circular tube or flue be subjected to external normal 
pressure., such as that of steam or water, the material of which 
it is made will be subjected to compression around the tube, in 
a plane normal to its axis. If the following notation be 
adopted : 

/ = length of tube ; 

d = diameter of tube ; 

/ = thickness of wall of the tube ; 

/ = intensity of excess of external pressure over internal, 

then will any longitudinal section //, of one side of the tube, be 
subjected to the pressure ^ — -. But let a unit only of length 

of tube be considered. This portion of the tube is approxi- 
mately in the condition of a column whose length and cross 
section, respectively, are Ttd and f. 

The ultimate resistance of such a column is, Art. 25 : 






As this ideal column is of rectangular section : 



656 COLLAPSE OF FLUES. [Art. 77. 



1= — 
12 



and 

P 

But F = pd, hence : 







(>) 



is the greatest intensity of external pressure which the tube 
can carry. But the formulae of Art. 25 are not strictly applicable 
to this ideal column. The curvature on the one hand and the 
pressure on the other tend to keep it in position long after it 
would fail as a column without lateral support. Hence, p will 
vary inversely as some power of d much less than the third. 

Again, it is clear that a very long tube will be much more 
apt to collapse at its middle portion than a short one, as the latter 
will derive more support from the end attachments ; and this 
result has been established by many experiments. Hence,/ 
must be considered as some inverse function of the length /. 

Eq. (i), therefore, can only be taken as typical in form, and 
as showing in a general way, only, how the variable elements 
enter the value of/. \i x, y and Zy therefore, are variable ex- 
ponents to be determined by experiment, there may be written : 

^ = '-M' ^^) 

in which c is an empirical coefficient. 

Sir Wm. Fairbairn (" Useful Information for Engineers, 
Second Series ") made many experiments on wrought-iron tubes 
with lap and butt joints single riveted. He inferred from his 



Art. 'jyP^ COLLAPSE of flues. 657 

tests that ^ = ^ = i. Two different experiments would then 
give: 

pld — cf" (3) 

p'l'd' = ct''' (4) 

Hence, 

log {pld) = log c -{- X log t ; 

log {p'l'd') = log c -\- X log t' \ 

in which ^^ log'' means '* logarithm." Subtracting one of these 
last equations from the other, the value of x becomes : 



log {^^ 

u'd') _ ^ Kpi'd'J 



^ ^ log (pld) - log {p'l'd') ^ Kpl'd'J 

log t- log t' ^^^It^ ' ' ^'-' 



As /, /, dj t, p'y /', d' and /' are known numerical quantities 
in every pair of tests, x can at once be computed by Eq. (5) ; 
c then immediately results from either Eq. (3) or Eq. (4). By 
the application of these equations to his experimental data, 
Fairbairn found for wrought-iron tubes : 

/ = 9^675,600 ^' (6) 

in which / is in pounds per square inch, while /, /and </ are in 
inches. Eq. (6) is only to be applied to lengths between 18 and 
120 inches. 

He also found that the following formula gave results agree- 
ing more nearly with those of experiment, though it is less 

simple : 

42 



658 COLLAPSE OF FLUES. [Art. 77. 

/ = 9>^75,6oo -^ - 0.C02 - (7) 

Fairbairn found that by encircling the tubes with stiff rings 
he increased their resistance to collapse. /;/ cases where such 
rings exist, it is only necessary to take for I the distance between 
two adjacent ones. 

In 1875 Prof. Unwin, who was Fairbairn's assistant in his 
experimental work, established formulae with other exponents 
and coefficients ('* Proc. Inst, of Civ. Engrs.," Vol. XLVL). 
He considered x, y and z variable, and found for tubes with a 
longitudinal lap joint : 

P = 7,363,000 ^-^^-, (8) 

From one tube with a longitudinal butt joint, he deduced : 

y2.2I 
/ = 9,614,000 ^--, (9) 

For five tubes with longitudinal and circumferential joints, 
he found : 

/= 15,547,000^^^, (10) 

By using these same experiments of Fairbairn, other writers 
have deduced other formulae, which, however, are of the same 
general form as those given above. It is probable that the 
following, which was deduced by J. W. Nystrom, will give 
more satisfactory results than any other : 



/ = 692,800^^ (11) 



Art. 78.] SOLID ROLLERS. 659 

At the same time, it has the great merit of more simple 
application. 

From one experiment on an elliptical tube, by Fairbairn, it 
would appear that the formulae just given can be approximately 
applied to such tubes by substituting for </, twice the radius of 
curvature of the elliptical section at either extremity of the 
smaller axis. If the greater diameter or axis of the ellipse is a, 

and the less b ; then, for </, there is to be substituted -y- . 

o 



Art. 78. — Approximate Treatment of Solid Metallic Rollers. 

An approximate expression for the resistance of a roller 
may easily be written. The approximation may be considered 
a loose one, but it furnishes a basis for an accurate empirical 
formula. 

The roller will be assumed tc be composed of indefinitely 
thin vertical slices parallel to its axis. It will also be assumed 
that the layers or slices act independently of each other, and, 
finally, that the material above the roller is of a thickness equal 
to its rad.ius. 

In Fig. I let AC = d. 

In Fig. I let DC = e. 

Let E' = coefficient of elasticity for the material over 

the roller. 
Let B = coefficient of elasticity for the material of the 

roller. 
Let w = intensity of pressure at A. 
Let P = total weight which the roller sustains per unit 

of length. 



65o 



CYLINDRICAL ROLLERS. 



[Art. JZ. 




Fig.l 



Let X and y be measured 
horizontally and vertically, 
respectively, from A as the 
origin. 

From Fig. I : 

BC = ^, AB ; and 



B'C = =r, A'B' . . (i) 



If / is the intensity of 
vertical pressure at any point, 



/fax = —rrr wax 
^ AB 



(2) 



But by Eq. (i) : 



A'B =. 



E 



E + E 



^A'C 



Also, if R is the radius of the roller : 



w 



A'C'=d-y; and AB = R ^ . 

E 



Hence, 



E E dP 

pdx — w . -^— (d — y) dx = — • . . (3) 

■^ E -\- E Rw ^ -"^ 2 ^^^ 



From the equation of the circle : 



y = R — ^/R^ - x' 



Art. 78.] CYLINDRICAL ROLLERS. 661 



Since 

P= 2 



dP, there results 



P = 



^ i^f^E') ('^ -^R-\-V2{R'- er e + y2R^ stn-^£) (4) 



Eq. (4) can be very much simplified for all ordinary cases. 
From what has preceded : 



When e is small : sin-'^ -— = —-; ^ = ^2Rd -\- d^ ; and 

(R' - cy^ = R--^^ nearly. 
Substituting in Eq. (4): 

^ ^ rWVe')^^^^ - '^''^ .... (5) 

Hence, as ^is small, nearly : 



p^R^f2W^^+^ (6) 



Or, for length / : 



P' = RI^/2W^^±^ (7) 



A simple expression for conical rollers may be obtained by 
using Eq. (6). 



662 



CONICAL ROLLERS, 



[Art. -jZ. 



As shown in Fig. 2, let z be the distance parallel to the 
axis of any section from the apex of the cone ; then consider a 




Fig.2 



portion of the conical roller whose length is dz. Let R-^ be the 
radius of the base. The radius of the section under consider- 
ation will then be : 



R^-^R.; 



and the weight it will sustain : 




E -^ E 
2w^ — , . zdz. 



Hence, 



P' = 



/'- 



a' 



'' a 



dP-^^^R. 



2W^ 



EE 



. (8) 



Eqs. (7) and (8) give, ultimate resistances if 7(/ is the ultimate 
intensity of resistance for the roller. 

It is to be observed that the assumption on which the in- 
vestigation is based leads to an error on the side of safety. 



Art. 79.] 



SPIKE DRIVING AND DRAWING. 



66^ 



Art. 79. — Resistance to Driving and Drawing Spikes. 

Some very interesting experiments on driving and drawing 
rail spikes were made by Mr. A. M. Wellington, C. E., and re- 
ported by him in the " R. R. Gazette," Dec. 17, 1880. He ex- 
perimented with wood both in the natural state and after it 
had been treated by the Thilmeny (sulphate of baryta) pre- 
serving process. 

*' The test blocks were reduced to a uniform thickness of 4.5 
inches ; this thickness being just sufficient to give a full bear- 
ing surface to the parallel sides of the spikes when driven to 
the usual depth, and to allow the point of the spike to project 
outwards. It was considered that the beveled point could add 

Spikes wej'e Standard : ^.^ inches x n; inch. 



KIND OF WOOD, 



Beech 

White oak, green 

Pin oak 

White ash 

White oak, well seasoned 

Black ash 

Elm 

Chestnut, green 

Soft maple 

Sycamore 

Hemlock 



NATURAL WOOD. 



To driving 
spike, pounds. 



Mean. 

5,953 



,433 



,090 



J 3,996) 
I 4,202 ) ^' 

]t;f43.«43 

2,910 



To pulling 
spike, pounds. 



Mean. 



^69 
,538 
6,469 



6,638 \ g^ 



553 
4,560 



281 



3,435) ^' 

r 3,260 

3.790 i -5' 

2,578*, 
3,645* ^' 
3,1^ 
3,iJ 



III 



f 3,188 
1,996 



PREPARED WOOD. 



To driving 
spike, pounds. 



Mean. 

7,283 ) 
7,656 f 7i4 



,472 



6,117 I 
4,589^5,353 

6.588 



5,978 



■6,283 



4,453 
4,453 
4,301 
3,380 J 
4,453 L 
4,148 P' 



4,147 



300 






To pulling 
spike, pounds. 



Mean. 

i:2g[«'42o 

(Split.) 



3-34o'l 
3,028 I 
1:300 3,290 

3,493 J 

'♦'''^M 4,175 
4,202 ) ^' '^ 



^'725U,877 
5,030 ) ' ' ' 

1,968 



664 



SHEARING BY BOLTS AND KEYS. 



[Art. 80. 



very little to the holding power of the spike, and it was desired 
to press the spike out again by direct pressure after turning the 
block over. . . ." 

The forces exerted in pulling and driving the spikes were 
produced by a lever. A few tests with a hydraulic press 
showed that the friction of the plunger varied from about 6 to 
18 per cent. 

The accompanying table gives the results of the experi- 
ments. 



Art. 80. — Shearing Resistance of Timber behind Bolt or Mortise 

Holes. 

Col. T. T. S. Laidley, U.S.A., made some tests during 1881 
at the United States Arsenal, Watertown, Mass., on the resist- 
ance offered by timber to the shearing out of bolts or keys, 
when the force is exerted parallel to the fibres. 




Fig.l 



Fig.2 



The test specimens are shown in Figs. I and 2. Wrought- 
iron bolts and square wrought-iron keys were used. All the 
timber specimens were six inches wide and two inches thick. 
The diameter of the bolts used (Fig. i) was one inch for all the 
specimens. The keys were i" x 1.5" and 1.125" x 1.5" as 
shown in Fig. 2. In all the latter specimens, failure took place 
in front of the smaller key where the pressure was greatest. 

In many cases the specimen sheared and split simultane- 
ously in front of the hole. By putting bolts through the 
pieces in a direction normal to the force exerted, so as to pre- 



Art. 8 1.] 



BULGING OF PLATES. 



665 



vent splitting, the resistance was found (in most cases) to be 
considerably, though irregularly increased. 



KIND OF WOOD. 



Spruce (bolts). 



White pine (bolts) 



Yellow pine (bolts) 



Yellow pine (square keys) . 



White pine (square keys) . 



Spruce (square keys) , 



CENTRE OF 
HOLE FKO.M 

END OF 
SPECIMEN. 



Inches. 

11 



TOTAL AREA 
OF SHEARING. 



Sq. inches. 

\i 

(. 32 

;i 

\ 28 
8 
16 
24 
28 
8 
16 
24 



ULTIMATE SHEARING RESISTANCE PER 
SQUARE INCH, IN POUNDS. 



(not thorough\y seasoned.) 
(wet timber). 



Unless otherwise stated, the wood was thoroughly sea- 
soned. 

The accompanying table gives the results of Col. Laidley's 
tests. 



Art. 81.— Bulging of Plates. 

A plate offers resistance to "bulging" when it is simply 
supported, or firmly fixed, around its entire edge, and carries 
a single, or uniformly distributed load acting normal to its 
surface. The very complicated nature of the stresses and 
strains existing in a plate thus acted upon, together with the 
fact that its conditions just before rupture are entirely different 



^^ BULGING OF PLATES. [Art. 8 1. 

from those accompanying the initial loading, give to the prob- 
lem a character of unusual intricacy, and, indeed, preclude a 
solution possessing a degree of approximation commonly ob- 
tained in questions relating to the elasticity and resistance of 
materials. 

An elegant analysis of the problem, considered as one of 
pure elasticity, may be found in '' Die Theorie der Elasticitat 
Fester Korper," by Clebsch. It is, however, of little value in 
connection with questions of ultimate resistance. 

The following roughly approximate, but simple, analysis may 
be used to suggest the form of an empirical formula which can 
be completed by the aid of experiments. 

Let the length, breadth and thickness of a rectangular 
plate simply supported around its edges, be represented by a, 
b and /, respectively, and let it first be loaded by a uniformly 
distributed pressure whose intensity (per unit of aU) is w. 

If the plate is supposed to consist of two sets of small 
strips or beams parallel to a and b^ those crossing each other 
at the centre must have the same deflection at middle. If, 
further, the uniform load iv be supposed to be so divided into 
two parts, w^ and zv\ that they would cause two rectangular 
beams whose spans are a and b to have the same centre deflec- 
tion, the following equation (see Eq. (26) of Art. 24) must 
obtain : 

"^w^a^ __ ^w'b* 

Then, since w' -\- w^ =^ «/, there must result : 
w = T ; and IV, — ~ . 

^4 4- ^4 ' ^ a^ _|_ /;4 

The bending moments at the centres of such beams would 
be (Eq. (27), Art. 24, and Eq. (14), Art. 18) : 



Art. 8 1.] FORMULM. 66y 

w^a^ 2KJ , wb^ 2K'I 

_^_; and -g---^. 



Since the beams are rectangular in section, / = — , 

12 

Hence : 

a; = 3!^^; and AT' = ^^. 

According to these hypothetical conditions the greatest in- 
tensity of stress at the centre of the plate will have the value : 

\ 2 J ^tW2 {a^ -\-b^) ' ' * 



Hence : 



^^^Y_j^^_y (2) 



O 



For square plates, a = b. 
Hence 



^ = ^g^; and / = 0.615^./-^ ... (3) 



If the edges are fixed, the greatest bending will occur along 
those lines ; and for K^ and K' then are to be put Y^K^ and 
VlK'. 

Hence : 

A I = 2 . -. — ; and K = 2 . -. , — ^ . (4) 

Since the greatest bending occurs along the edges, these 



668 buIging of plates. [Art. 8i. 

are the expressions for the greatest intensities of stress. If ab^ 
is greater than a'b, then is K^^ greater than K' ; and vice versa. 
In the first case the expression for / is : 



/ = o.707ab^ ^ ^^^ , ^^, ^ .... (5) 



w 



But if K' > K„ or, a'b > ab' : 



i = o.707a^b./^-—---^ .... (6) 



{a^ + b^) K 
If the plate is square : 



K=-,; and, t = -^ ^ ■ • • • (/) 

If a plate is loaded with a single weight P, it may be sup- 
posed to be divided in the same manner as z£/ ; so that 



P^J^ P' = P, 

The equation of middle deflections for ends simply sup- 
ported then becomes : 

P,a^ P'b' 



Hence : 

', a^P , p b^P 



^3 + ^3 ' ' ^ ^3 _j_ ^3 • 

Proceeding in precisely the same manner as before: 



Art. 8 1.] FORMULA. 669 



abP 



and 



^ = ^-^^ t' (^3 _|_ ^3j V^^ + ^ .... (8) 

t = 1.03 ^-(^3H-^3) v^* + ^^; . . . . (9) 



If the plate is square : 



^=075^; and, ^=0-S7a/^ . • (10) 



7/" t/ie edges are fixed in position^ the hypothetical beams 
are fixed at each end and loaded at the centre, and the greatest 
bending moments (at centre and ends alike) are thereby re- 
duced to one-half their preceding values, or, what is the same 
thing, 2/^ is to take the place of f in Eqs. (8), (9) and (10). 

Hence : 

abP , , ^ 

-^= Q-53 f{a'J^ b^) V^' + ^' (lO 

/ abP , \'^ , , 

t = 0.73 U (^3 -I- ^3) Va^ + ^V • • • • (12) 

If the plate is square : 

^ = 0.375^; and ^ = 0.613 W^ . . (13) 

These equations are of little value as they stand, excef)t as 
indicating a form of formula to which empirical coefficients are 
to be fitted. The hypothetical division of the plate into small 
^beams is very far indeed from being correct. In the empirical 
determinations which follow, therefore, K will not be the 



6/0 BULGING OF PLATES. [Art. 8 1. 

greatest intensity of stress in the plate, but a coefficient or 
quantity partly analytical and partly experimental. 

Circular plates have not been considered, because square 
ones furnish the requisite type of formula. 

Experiments have thus far been made on square and circu- 
lar plates only ; hence, oblong rectangular plates will not again 
be noticed. 

Kirkaldy's experiments on Fagersta steel plates and Fair- 
bairn's on wrought-iron ones would seem to indicate that the 
thickness t varies about as {wf''^ or [Py-^ ; but the variation in 
diameter or side of square was not sufficient to establish any 
relation between / and «, while other elements remain the 
same. Regarding, therefore, K as an empirical quantity which 
may have different values for square and circular plates, Eqs, 
(3), (7), (10) and (13), may be written as follows : 

and t — o.6iKa — ^= . . (14) 
VAT 



and / = OAa- — = . . . (15) 



K 


8/^ 


K 


— - tt/ 
4/^ 


K 


_ 3^ p,.6 

At' 


K 


_ 3^^ pi.6 

8/^ 



and / = 0.87 ^''_. . . . (16) 



and / — o.oit r=r- . . (17) 



iftrkaldy made twenty experiments with mild Fagersta 
steel circular plates, 12 inches in diameter. He forced these 
through an aperture 10 inches in diameter by the pressure of a, 
very blunt point. The edge of the aperture on which the 
plate rested was rounded ; hence the initial diameter of aper- 



Art. 8 1.] STEEL CIRCULAR PLATES. 6/1 

ture was somewhat more than lo inches. Eqs. (i6) are the 
ones to be used in connection with these experiments. 

From the first member of that equation, K was computed 
for a number of different experiments, by substituting the 
numerical values of /*, / and a. In this manner the following 
values were found to give good results : 

For unannealed mild Fagcrsta steel circular plates : 

K = 6,760,000,000. 
Hence : 

/ = 0.000,010,6 a/^zP°^ (18) 

For annealed mild Fagersta steel circular plates : 

K — 5,710,000,000. 



Hence : 



Eq. (16) gives : 



/ = 0.000,011,52 y^,7po.8 ^j^) 



P = /^vz Y'^ . 

\0-^7VaJ ' 

Table I. contains the results of computation by this formula 
and those obtained in the tests. On account of the rounded 
edge of the supporting ring, K was so taken that P, as com- 
puted, is a little larger than its experimental value. None 
of these plates were cracked, but they were bulged at the cen- 
tre from 3.00 to 3.45 inches. 

In ** Engineering" for Sept. 28, 1877, Robert Wilson de- 
scribes four experiments on unstayed flat boiler heads sub- 
jected to hydraulic pressure. These flat circular plates were 



6/2 



BULGING OF PLATES. 



[Art. 8 1. 



TABLE I. 

Circular Plates simply Supported. 



UNANNEALED. 


ANNEALED. 


/, in inches. 


/*, in pounds. 


/", in inches. 


P, in pounds. 


Experimental. 


By /ortnula. 


Experimental. 


By /orjnula. 


» 

t 

1. 

4 


215,690 

162,740 

104,850 

71,800 

35,400 


219,420 

166,000 

115,860 

69,800 

29.350 


5. 

8 

i 

1 
4 

\ 


198,000 

154,330 

95,600 

59,430 
25,430 


196,530 

148,690 

103,780 

62,520 

26,290 



Each ^^ experitnentaP^ result is a mean of two. 

riveted to angles encircling the body of the boiler. The edges 
of the plates were thus fixed, and Eqs. (15) are therefore to be 
used. Proceeding in precisely the same manner as before, the 
following values were established : 

For wrought-iro7t flat boiler heads y with fixed edges : 



K = 11,000,000. 



Hence : 



t — 0.000,01 5 rt'TC/' 



cr,0.8 



(20) 



w was taken in pounds per square inch ; it has the value ; 



^ = (^K_y, 



Art. 8 1.] WROUGHT-IRON PLATES. 673 

The results of the experiments, and of this formula, are : 

DIAMETER, t, ^7W, IN POUNDS PER SQ. IN.—, 

INCHES. INCH. Experimental. By formula. 

34-5 h 280 349 

34 5 » 200 211 

26.25 t 371 296 

28.25 f 300 270 

The agreement, in this case, is not satisfactory. It is prob- 
ably due to the lack of a proper exponent of a. These plates 
were fractured along the lines of rivet holes in the edges. 

Two means of four experiments by Fairbairn remain to be 
considered. His plates were square ones of wrought iron, 
firmly fixed to a square frame 12 inches by 12 inches in the 
aperture. The force was applied by a blunt point at the 
centre, consequently Eqs. (17) are to be used. 

By precisely the same method already used, the following 
results were established : 

For wrought-iron \2-inch square plates^ with edges firmly 
fixed : 

K = 390,000,000. 

t = 0.000,031 Va P°^ (21) 

The expression for the indenting force is : 

I /sk\ 

P = 




The experiments and computations are : 

DIAMETER, i, , P, IN POUNDS. > 

INCHES. INCH. Experimental. By formula. 

12 \ 16,780 16,350 

12 \ 37,720 38,890 

The plates gave way at the centre, under the blunt point. 
43 



674 



SPECIAL CASES OF FLEXURE. 



[Art. 82. 



Some experiments by Kirkaldy, in 1875, on wrongJit-iron 
circular plates simply supported around the edge, show that for 
12-inch plates forced through a lo-inch aperture with rounded 
edge, there may be safely taken : 

/— 0.000013 V^ -^°-^ (22) 

In all the preceding formulae, a and t are to be taken in 
inches ; w in pounds per square inch, and P in pounds. 

The investigations can only be considered provisional. Al- 
though they give, as a whole, tolerably satisfactory results, the 
range of the experiments is far too small for the establishment 
of thoroughly reliable formulae. Experiments on which a 
proper exponent of a can be based, are yet wholly lacking ; and 
as the only resort, that found in the rough analysis has been 
retained. 

Art. 82. — Special Cases of Flexure. 

There are a few cases of flexure which, while not frequently 
found in engineering experience, are of some practical impor- 
tance, and are occasionally required. The two or three which 
follow involve the integration of some linear differential equa- 
tions that are treated in all the advanced works on the integral 
calculus ; consequently the operations of integration will not 
be given here, but the general integrals will be assumed. 



Flexure by Oblique Forces 
In Fig. I let OX represent 




Fig.l 



beam acted upon by the 
oblique forces P, which 
make angles a with the 
axis of X. The origin 
is supposed to be taken 
anywhere on the axis of 
the beam. If right-hand 



Art. 82.] SPECIAL CASES OF FLEXURE. 675 

moments are positive and left-hand negative, the component 
P sin a will have the negative moment — P sin ax about O, 
The lever arm of P cos a^ if the deflection w is positive, is 
-f- Wj and its moment P cos a ,w is positive. Hence the result- 
ant moment of any force, P, in reference to the origin O is : 

EI —. — = — P sin a , X -\- P cos a , w . . ( i ) 
dx"^ 

If a is greater than 90°, cos a is negative, so that if 

. P cos a . rr P sin a . x 

A — ± — ^^-y— and F = ^^7 , 

EI EI 

the two cases may be expressed by the equation : 

^'^J^Aw=V .^ . (2) 

dx" ^ ^ ^ 



If a = -{- V — A, and b—— V — Ay the general integral 
of Eq. (2) is : 



^ix 



w = Ce'''' + C'e^'' H ^^ I Ve- '^^ dx ^^ — -j Ve- *^ dx ; (3) 

a — ] a — b) 



in which C and C are arbitrary constants to be determined by 
the special conditions of any given problem, and e = 2.71828. 
When a is less than 90° and 



_ /P cos a A r= — /^ ^^^ ^ 

^ '~Er~ ' V ~~EI~ 



Eq. (3) becomes : 

w = Cc^"" + C7V*^ -{- X tan a -{- C, . . . . (4) 



^'J^ SPECIAL CASES OF FLEXURE. [Art. 82. 

6^1 is another arbitrary constant to be determined by the par- 
ticular circumstances of a given case. 

The conditions on which the determination of these con- 
stants rest are expressed by giving known values to w and — — 

for values of x, also known. 
If a is greater than 90° : 




P cos a J 

— - — . V— I 
EI 



and Eq. (3) becomes : 

zv = C cos i — ^ — \ X -\- C sin [ — — — 

X — tan a . X -{- C^ (5) 

As before, C, C and C^ are to be determined by the cir- 
cumstances of each case to which the equation is applied ; and 
the value of cos a, it is to be remembered, is to be substituted 
with the positive sign. 

Let a column with one fixed and one free end, and with the 
force P acting parallel to its original axis, be considered. Since 
a= 180°: 



w — C cos [ \ -^j- ' ,r ) H- C sin [ \/ -^r ' x \ -\- C^ (6) 





Let the origin of co-ordinates be taken at the free end. 
Now since w will vary from o to its greatest value at the fixed 
end, there can be no portion of it which is constant. Hence, 
Cj^ must be equal to zero. Also, w must equal zero for x = o. 
This condition gives (7=0. 



Art. 82. J COLUMNS. ^yy 

The value of w then becomes : 



w = C sin [ \l ^j ' X \ (7) 



.-. 2""^ V^'''(v^ * ""^ • • • (8) 



But if / is the length of the column, — — =0 for ;ir = /. 
Hence : 

EI 



<^0S f \ IpT '^1=0; 



or if n is any whole number from o to infinity : 



^ . / = y^{2n + i)7r (9) 



If the value of . / be taken from Eq. (9) and inserted 

in Eq. (8), there will result : 

dw ^, IP ( x{2n ■\- \)\ . , 

Eq. (10) shows that for values of x equal to I, 3, 5, 7 . . . 

times ■ , — r— = o. The most dangferous supposition, 

2;^ + I ' dx ^ ^^ 

i. c, that which requires the greatest value of P, \s n = o. 



(>'j'^ SPECIAL CASES OF FLEXURE. [Art. 82. 

This value of n in Eq. (9) gives : 

(") 




The ultimate resistance of the column is thus seen to be 
independent of the deflection, as was found for a different case 
in Art. 25. The end of the column, in this case, which carries 
the load is free to deflect laterally, but in Art. 25 both ends 
were supposed to be fixed in a lateral direction in reference to 
each other. In the latter case the resistance is seen to be nine 
times as great as in the present. 

Since : 



cos \ A / TH- ' / = O, Sill \ / T^ • / z= I . 





Hence, if w^ is the deflection of the free end from a vertical 
tangent to the fixed, Eq. (7) becomes, for x == I : 

.w^ = C, 
In general, therefore : 



w ^ w^ sin 



For the same value of x, therefore, w varies directly as w^y 
and the relative deflections may be computed by the equa- 
tion : 

w . /(2n + i)7tx\ , , 




Art. 82.] FLEXURE B V NORMAL LOAD. 679 

or in the ordinary case : 

w . nx . , 

^r'"'^ ^'^^ 

Eq. (i) was written for one force only. If any number of 
forces act : 

EI —7^ =z ^( — P sin a , X -\- P cos a . w) ; 

dx^ 



and in place of w there is to be put ^w. 



General Flexure by Continuous Normal Load. 

The most general case of flexure by a continuous normal 
load, is that in which the intensity (load per unit of length of 
beam) is a variable quantity. Let x be an abscissa measured 
along the original axis of the beam, and let w represent the 
deflection. Then the intensity of the load may be represented 
hy f{x^ zv). It was shown in Art. 20 that : 

—7 — = EI —y— = p = fix, w) . . . . (15) 
dx^ dx^ ^ -^^ ' ^ ^ ^^ 



The integration of the equation : 

d^zu _ /(x, zv) 
'd^ ZT^ ' 

will depend upon the form of the function /(;r, zv). 

Let it be supposed that f{x, zv) = ex, c being a constant. 
Then if A, A^, A^ and A^ are constants of integration, there 
will result : 



680 SPECIAL CASES OF FLEXURE. [Art 82. 

120 is/ 6 2 1213 \ J 

Again, if /"(-f, ^) = cw^ c, as before, being a constant : 



dx^ EI 

For simplicity of notation, let : 



EI ' 



• e» ♦• • ill ^ J 



then the general integral of Eq. (17) becomes : 

w = Ae^'' + A^e-"'' + A^ cos ax + A^ sin ax . . (18) 

In Eq. (18) ^ = 2.71828 is the base of the Naperian log- 
arithms ; while in both Eqs. (16) and (18) A^ A^, A^ and A^ are 
arbitrary constants to be determined by the circumstances of 
each individual case. 



CHAPTER XII. 

Working Stresses and Safety Factors. 

Art. 83.— Definitions. 

In all metallic and timber constructions the greatest (sup- 
posed) possible loads are determined from the attendant cir- 
cumstances of the different cases, and then the stresses induced 
by these greatest loads are computed. These stresses are 
called the " zuorking stresses,'' 

The ultimate resistance of any piece in a structure divided 
by the working stress gives a number called the ^^ safety factor,'' 
Occasionally the reciprocal of this number is called the safety 
factor, though but seldom. 

The intensity of the ultimate resistance of any piece in a 
structure divided by the intensity of the working stress, will also 
give the safety factor. This is the more usual and convenient 
form, since it does not involve the cross section of the piece. 

The values of safety factors depend upon many circum- 
stances, such as kind and character of material, kind of stress, 
circumstances in which material is used and the amount of 
variation of stress in the piece, or the fatigue of the material. 
The safety factor is intended, also, to cover both computed 
stresses and others which are recognized, but are not within the 
reach of exact analysis. The latest practice among American 
engineers will be illustrated in the following Articles by ex- 
tracts from specifications drawn for some first-class construc- 
tions. 



682 specifications: [Art. 84. 

Art. 84. — Specifications for Sabula Bridge. 

The following extracts are from the specifications for a 
bridge at Sabula, on the line of the Chicago, Milwaukee and 
St. Paul Railway. 

*' Qiiqliiy of Iron and Steel. — All eye bars, rods, bolts, and 
pins shall be made of a tough, ductile, fibrous iron, uniform in 
quality, and which shall be capable of withstanding the follow- 
ing tests when applied to full-sized sections of the material 
tested. 

Round bars up to ly^ inches in diameter must bend double, 
or until inner sides are in contact, when cold, without showing 
signs of fracture. 

Square bars must bend cold through 180 degrees around a 
cylinder having a diameter equal to two-thirds the length of 
side, without showing signs of fracture. 

Flats must bend cold through 180 degrees around a cylinder 
having a diameter equal to the length of the shortest side, 
without sign of fracture. 

The ultimate strength of the bar iron used shall not be less 

^, /. 7,000 X area \ , . , 

than ( 52,000 — ^ ) pounds per square mch ; area 

and periphery being expressed in imches. 

The elastic limit shall not be less than 26,000 pounds per 
square inch, and the elongation of the bar before rupture shall 
not be less than 15 per cent, in 12 diameters. 

The reduction of area at breaking point shall not be less 
than 25 per cent, of the original section. 

All plate and shape irons used in tension members, or in 
members exposed to both compressive and tensile strains, shall 
fulfill all the foregoing conditions when tested in specimens of 
one inch area and 15 inches length of smallest section, except 
that the breaking strain per square inch shall not be less than 
50,000 pounds for angles, 49,000 pounds for beams and channel 
iron, and 48,000 pounds for plate iron. 



Art. 84.] SAB U LA BRIDGE. 683 

Iron for compression members must be tough, fibrous, uni- 
form in quality, and with an elastic limit of not less than 25,000 
pounds per square inch. 

• ••••••• 

All cast iron shall be good, tough, gray iron of such quality 
that a bar five feet long, one inch square, and 4^ feet between 
knife-edge supports will sustain a weight of 500 pounds on 
knife edge at middle of beam before breaking. 

All steel shall be of a good quality of mild steel, having an 
ultinrate tensile strength of 90,000 pounds, or over, per square 
inch, an elastic limit of not less than 45,000 pounds, a ductility 
of 12 per cent, in 12 diameters, and not less than 15 per cent, 
reduction of area at breaking point. 

Specimens one square inch in area of section shall bend 
cold through 180 degrees around a cylinder whose diameter is 
4 times the length of the shortest side of the test piece. 

All bar and rod iron shall be tested in full-sized sections 
whenever practicable. ^ 

All tension iron shall be rolled from piles composed of 
piling pieces, each the full length of the pile. The use of old 
rails wAW not be allowed in the piles for this grade of iron. 

• ••••••• 

Eye Bars. — ....... 

Bars of the same class and belonging to the same panel 
shall be drilled at the same temperature. 

No error in length of bar or diameter of pin hole exceeding 
■^-^ inch will be allowed. 

The section of metal opposite the centre of pin hole, across 
the eye, shall be proportioned according to the following table, 
the diameter of the bar being the unit. 



684 



SPECIFICA TIONS. 



[Art. 84. 



PIN. 


BAR. 


EYE SECTION. 


Upset heads on weldless bar. 


Heads rolled on bars. 


0.67 

0.75 

I.OO 

1.25 

1-33 

1.50 

1-75 
2.00 


I.O 
I.O 

1.0 

I.O 
I.O 
I.O 
I.O 

I.O 


1.50 
1.50 

1.50 
1.60 
1.70 

1. 85 
2.00 
2.20 


1-33 
1-33 
1.50 
1.50 
1.60 
1.67 
1.67 
1-75 



Pins. — Pins must be turned 



no error of more than -^^ inch in diameter being 



allowed. 



Pins connecting laterals with other members shall be turned 
down to a diameter of not more than -^-^ inch less than the pin 
holes. 



Rods. — ....... 

. . . Screw ends shall be upset so as to give 10 per cent, 
more sectional area at the bottom of the screw thread than in 
the body of the bar. . . . 

Chords of Pivot Span. — ..... 

. . . No error exceeding -^^ inch in length of part or in 
diameter or position of pin hole will be allowed. The pin holes 
may be bored -^-^ inch larger than the pin, but this is the utmost 



Art. 84.] SABULA BRIDGE. 685 

limit. Rivet holes in the splices and in all steel plates shall 
be punched for ^ inch rivets and then reamed for l/^ inch 
rivets. ....... 

Posts of Pivot and Fixed Spans. — .... 



When necessary, pin holes in posts, chords or tie struts 
shall be reinforced by additional material, which must contain 
rivets enough to transmit the strain to the original member. 
The open sides of posts, chords, struts and tie struts shall be 
connected by lattice or trellis bars, the angles of which shall 
not exceed 63° 25' for single bars or 45° for double bars with 
riveted intersections. 

The unsupported length of any lattice bar shall not exceed 
50 times its thickness ...... 

Riveted Work. — ..... 

Rivet holes shall not be spaced less than 2j4 diameters 
between centres, nor more than 16 times the thickness of the 
thinnest outside plate apart, 9 inches being the maximum pitch 
allowed in plate riveting. 

No rivet holes shall be less than i^ diameters from the end 
of a plate, or i^ diameters from the side of a plate, nor ever 
less than i ^ inches from centre of hole to edge of plate, ex- 
cept in cases where the plate or side of angle is less than 2^ 
inches. 

The diameter of hole shall not exceed the diameter of rivet 
more than -f-^ inch. 

When two or more thicknesses of plate are riveted together, 
the outer row of rivets shall, if practicable, not exceed three 
rivet diameters from the edge of the plate. 

Where plates more than 12 inches wide are used in the 
compression flanges of girders or floor beams, an extra line of 
rivets, with a pitch of not over 9 inches, shall be driven along 
each edge to draw the plates together. 



686 SPECIFICATIONS. [Art. 85. 

Turn Table. — This will be rim-bearing, 

• ••••• 

The circular girder will be of wrought iron 26 feet in diam- 
eter, and three feet in depth, and will be made of the grade of 
iron prescribed for material in tension. The wheels will be 18 
inches in diameter and 8 inches face, and will be turned to a 
uniform size and bored. (The draw or pivot span is 360 feet 
long.) 

• •••••• • 

C. Shaler Smith, D. J. Whittemore, 

Consulting Engmeer. Chief Engineer. 

Chicago, Milwaukee & St. Paul Railway.'* 

These specifications, complete, were printed in the " Ameri- 
can Engineer," for August, 1881. 

Art. 85. — Specifications for Albany and Greenbush Bridge. 

These specifications were prepared by Alfred P. Boiler, C. E., 
chief engineer of the Albany and Greenbush Bridge Co. This 
bridge crosses the Hudson River from Albany to Greenbush, 
N. Y., and is built entirely of wrought iron, with the exceptions 
of some details and the timber of the flooring. 

The following are the portions of the specifications bearing 
upon the subject under consideration : 

"...., the maximum stresses shall be as follows : 

Tension. 

Floor beam hangers 6,500 lbs. per sq, inch. 

Counter braces 8,000 ** " " " 

Main braces, except first three end panels 10,000 " " " " 

Lower chord and upper chord, when of eye bars, also first 

three main braces at ends 11,000 " " " " 

Lower chord when of channel bars, taking net section 

only 10,000 *' " " " 

Wind pressure stresses 15,000 ** " " ** 



Art. 85.] GREENE USH BRIDGE. 6Sy 



Compression Members. 

Proportioned according to Gordon's formula, with the fol- 
lowing constants in the numerator : 

For Phoenix columns 11,000 lbs. per sq. inch. 

For channel and beam iron sections 10,000 



<< ({ i( 



Transverse Stresses. 

On pins 15,000 lbs. per sq. inch 

On rolled beams 10,000 " " " '* 

On riveted beams 8,000 " " " 

On wooden stringers 1,000 " " " " 

Pressure Stresses. 

On bearing surfaces of pins (bearing surface = diameter 

of pin X width of bar) 12,500 lbs. per sq. inch. 

On bearing surface of rivets 10,000 ''*' " " " 

On masonry 300 " " *' " 



Tests. — Full sized bars of flat, round or square iron, having 
a section of not over 4.5 square inches, must exhibit an ulti- 
mate resistance of 50,000 pounds per square inch, and stretch 
12.5 per cent, of their lengths. Bars of larger section than the 
above will be allowed a reduction of 1,000 pounds per square 
inch for each additional square inch of sectional area, down to 
a minimum of 46,000 pounds per square inch. Specimen pieces 
taken from bars, and having a uniform section of one square 
inch or less for a length of 10 inches, must exhibit the follow- 
ing minimum values : 

From bars of 4.5 square inches in section and under, an 
ultimate resistance of 52,000 pounds per square inch with a 
stretch of 18 per cent, in 8 inches. 

From bars of over 4.5 square inches in section a reduction 
of 500 pounds per square inch for each additional square inch 



688 SPECIFICATIONS. [Art. 86. 

of section down to a minimum of 50,000 pounds per square 
inch. 

Specimens from angles, beams, channels or plates must 
show an ultimate resistance of 50,000 pounds per square inch 
with 15 per cent, elongation in 8 inches. 

All iron for tension members, whether bars, angles or plates, 
must permit of being bent cold, without cracking, on a diam- 
eter not greater than twice the thickness of the bar, plate or 
angle ; the cold-bend test on angles to be made after the two 
legs are severed. Any of the above classes of iron, when 
nicked and broken, must exhibit a fibrous fracture, almost 
entirely free from crystalline specks. 

Turn Table. — ...... 

. . . Centre bearing will be a flat pin, proportioned to 
a pressure of 6,000 pounds per square inch, with two steel 
(finally made of phosphor bronze) discs, the whole with proper 
provision for oiling. 



Art. 86. — Niagara Suspension Bridge. 

In his " Report on the Renewal of the Niagara Suspension 
Bridge," Mr. Leffert L. Buck, C. E., has given some data and 
calculations, from which he deduces that the safety factor for 
the cables is : 

11,000 H- (1,400 X 1.78) = 4.41. 

The total load between the towers being 1,400 tons and 
the ultimate resistance of the four wrought-iron cables, 11,000 
tons, while 1.78 is the ratio between the cable tension at the 
top of the towers and the vertical load between the towers. 

The new iron and steel stiffening truss is designed for a 
safety factor of 5. 



Art. Sy.] MENOMONEE BRIDGE. 689 

In the towers, built of limestone, he found a safety factor 
of nearly 23. 



Art. 87. — Menomonee Draw-Bridge. 

The specifications for this wrought-iron bridge (located on 
the Chicago, Milwaukee and St. Paul Railway) were written by 
C. Shaler Smith, C. E., and published in full in the "Ameri- 
can Engineer," for May, 1881. 

The following are portions of these specifications : 



Quality of h'on. — The iron subject to tensile strain shall be 
tough, ductile, and of uniform quality, capable of sustaining 
not less than 50,000 pounds per square inch of sectional area 
when tested in large and long lengths, to have an elastic limit 
of not less than 26,000 pounds per square inch ; . . . 



Tensile Members. — The tensile members shall be so propor- 
tioned that the maximum stresses produced by the weight 
of the structure, and the specified moving and engine load 
shall in no instance exceed the following : 

LBS. PER 
SQ. JNCH. 

For tensile stresses in primary members or those vipon ^vhich the principal 

weight comes directly from the floor beam 8,000 

For stresses in secondary members, or those which receive their principal 

stresses through the primaries 9,000 

Stresses in tertiary members 10,000 

Stresses in end suspenders 8,000 

Stresses in common floor-beam suspenders 4,000 

The foregoing are the stresses to be used where the speci- 
fied members are eye bars or bolts, but if they consist of 
riveted sections the' stresses allowed shall be as follows : 
44 



690 SPECIFICATIONS. [Art. 8/. 

LBS. PER 
SQ. INCH. 

For splice plates in tension 7,000 

For riveted members in tension in chords 8,500 

For riveted members in tension in web 8,000 

When any member is exposed to stresses in opposite 
directions, the sections shall be determined by the following 
formula, in which 5 represents the sectional area in square 
inches required, and the '' column strength per square inch " 
of sectional area is determined by the formulae hereinafter 
given : 

^ _ maximum tension maximum compression 



10,000 column strength per sq. inch 



Provided that the section thus formed shall not be less 
than required by the foregoing specifications for members in 
tension, or the following specifications for members in com- 
pression. 

(For these formulae see Art. 50, page 448.) 

Crippling Stresses. — The ultimate crippling resistance in 
pounds per square inch of section of the several forms of posts, 
struts and chords will be determined by the foregoing for- 
mulae, in which / -r- d or H equals the length between the end 
bearings in terms of the least diameter. 

The maximum stress permitted in any purely compressive 
member will be the quotient resulting from dividing the ulti- 
mate resistance, as determined by the above formulae, by a 
coefficient of safety (safety factor) equal to : 



4 + 



^^H,'' as before, being the measure of length in terms of least 
diameter. 




Art. 87.] MENOMONEE BRIDGE. 69 1 

Wind Strains. — ...... 

. . . shall be resisted by lateral and vertical rods propor- 
tioned to 15,000 pounds per square inch in tension, and lateral 
struts proportioned to a safety factor of four (4). 

Shearing and bending stresses at the lateral connections 
. . . shall be resisted by members so proportioned that the 
maximum shearing stresses shall not exceed 10,000 pounds per 
square inch, and the maximum flexure or bending stresses 
shall not exceed 22,500 pounds per square inch. 

Floor Beams and Track Stringers. — . 

. . . the resulting stresses (in floor beams) . . . shall 
not exceed 8,000 pounds per square inch in compression or 
10,000 pounds per square inch in tension. 

The stringers immediately under the rails shall be . . . 
subject to the same conditions as to limit of tensile and com- 
pressive stresses as specified above for floor beams. 

If the floor beams are of built sections, the rivets must be 
so spaced that between the points of application of the load 
and the points of support there are rivets enough to transmit 
the flange stresses to the web and from the web to the flange 
without exceeding a shearing stress of 7,500 pounds per square 
inch upon the rivets, or of 8,000 pounds per square inch of 
mean pressure on the semi-intrados of the rivet holes. 

Pins. — The shearing stress on any pin must not exceed 
7,500 pounds per square inch of its sectional area. The stress 
per square inch oxv extreme fibres, caused by bending, must 
not exceed 15,000 pounds, and in determining this bending 
stress the leveras^e distance shall be considered as from centre 
of eye bar to centre of bearing, or of opposite eye bar. 

No pin shall have a less diameter than two-thirds of the 
width of the widest bar coming upon it. 

The bearing surface of any pin on chord, tie or post shall 



692 SPECIFICATIONS. [Art. 87. 

not be exposed to a greater mean compressive stress than 
8,000 pounds per square inch. 

Riveted Work. — ...... 

All rivets with crooked heads or heads not formed centrally 
on the shank, or rivets which are loose either in the rivet hole 
or under the shoulder, shall be cut out and new ones put in 
their places. 

• ••• •••• 

The diameter of the rivet hole shall in no case exceed the 
diameter of the rivet by more than -f^ inch. 

In members consisting of two or more pieces of shape iron 
connected by lattice or lacing bars, there shall be connection 
plates at each end, the row of rivets in which shall be not less 
than one diameter of the member in length. ... in all 
riveted work the distance between rivet supports across the 
plate shall not exceed thirty (30) times the thickness of the 
plate, and no closed section shall have members of less thick- 
ness than -f^ inch. 

All rivets in splice or tension joints must be symmetrically 
arranged so that each half of a tension member or splice plate 
will have the same uncut area on each side of its centre line. 

• ••*.••• 

No rivet shall be exposed to more than 7,500 pounds per 
square inch in shear, or more than 8,000 pounds mean pressure 
per square inch of semi-intrados. 

Bed Plates and End Supports. — The bed plate supporting 
turn-table centre shall be so proportioned that the pressure on 
the masonry shall not exceed 25,000 pounds per square foot 
while the span is rotating, and the pressure of the wheel track 
and end supports of span on the masonry shall not exceed this 
limit when the bridge is fully loaded as heretofore specified. 

Turn Table. — . . , 



Art. 87.] MENOMONEE BRIDGE. 693 

If a wearing or friction centre pin is used, the pressure 
while rotating shall not exceed 6,000 pounds per square inch. 

If a lubricated centre pin is used, the weight while turning- 
must not exceed 1,000 pounds per square inch on the spindle 
bearing. 

If a Sellers' centre is used, the rotating load per lineal inch 
on the steel rollers shall not exceed that given by the follow- 
ing formula for steel rollers on steel plates : 



P = V 3, 07 2, 000^; 

in which formula P= pressure per lineal inch of roller, and d 
the mean diameter in inches. 

The load per lineal inch of face of wheel (for rim-bearing 
table), while the span is turning, shall not exceed that given by 
the formula : 



P — ^352,000^ ; 

for cast-iron wheels upon wrought-iron wheel track. 

For cast-iron wheels upon cast-iron track, the load per 
lineal inch of face of wheel, while the span is turning, shall not 
exceed that given by the formula : 



P = \/222,222d ; 

For steel wheels upon steel track, use the following formula 
as to limit of pressure upon lineal inch of wheel face : 



P = V 1, 296,000^. 
For steel wheels upon wrought-iron track 
P — ^/i ,024,000^ . 



694 SPECIFICATIONS. . [Art. 88. 

And for steel wheels upon cast-iron wheel track, use : 



• • •••••• 

Workmanship, Painting, etc, — . . 

The parts composing the posts or struts must be of entire 
lengths, without splicing between end bearings. 

• •••••• • 

All bars subject to tensile stress shall be tested by the con- 
tractor, . . . , to 18,000 pounds per square inch of sec- 
tional area." 

Art. 88. — Franklin Square Bridge. 

The specifications from which the following portions are 
selected, apply to the construction of a bridge over Franklin 
Square, New York City, on the line of the East River Bridge ; 
they were prepared by the Chief Engineer, Washington A. 
Roebling, C.E. 

.....a*. 

10. .AH members shall be so designed that the stresses 
coming upon them can be accurately calculated. 

12. For wrought iron the following clauses apply : 

I. Members in tension shall be so proportioned that the 

maximum stresses shall not cause greater tensions than the 

following : 

LBS. PER SQ. IN. 

On lateral bracing 15,000.00 

On solid rolled beams used as cross floor beams and stringers. . 10,000.00 

On bottom chords and main diagonals 10,000.00 

On counter rods and long verticals 8,000.00 

On bottom flanges of riveted cross girders, net section 8.000.00 

On members liable to sudden loading and shocks 6,000.00 



Art. 88.] FRANKLIN SQUARE BRIDGE. 695 

II. Members in compression shall be so proportioned that 
the maximum load shall not cause a greater compression than 
that determined by the formulae : 

D 8,000 , , 

F =: ' — J- , for square ends. 

1 + 



40,oooi^^ 



^ 8,000 , J J • J 

P =. — , for one square end and one pm end. 

1 + 



30,oooi^^ 
P z= '- — — , for pin bearing at each end. 

20,000i2^ 

P = allowed compression per sq. in. of cross section. 
/ = length of compression member in inches. 
R — least radius of gyration in inches. 

III. The lateral struts shall be proportioned by the above 
formulae to resist the resultant due to an initial stress pro- 
duced by adjusting the bridge, assumed at 10,000 pounds per 
sq. in. upon all rods attached to struts. 

IV. In beams and girders, compression shall be limited as 
follows : 

LBS. PER SQ. IN. 

In rolled floor beams used as cross floor beams and stringers. . . 10,000.00 

In riveted plate girders used as cross floor beams 6,000.00 

In any riveted girder under 20 feet long 5,000.00 

V. Members subject to alternate stresses of tension and 
compression shall be proportioned to resist each of them, but 
the sectional area of such members shall be increased if the 
engineer requires it. 



696 SPECIFICATIONS. • [Art. 88. 

VI. The diameter of a pin shall not be less than two-thirds 
of the largest dimension of any member attached to it, and its 
effective length shall not be greater than four times its diam- 
eter, plus the breadth of the foot of the post through which it 
passes. The shearing stress upon any pin shall not exceed 
7,500 pounds per sq. in. ; the crushing stress upon the pro- 
jected area of the semi-intrados (diameter multiplied by thick- 
ness of the piece) of any member connected to the pin, shall 
not exceed 15,000 pounds per sq. in., nor when the centres of 
the bearings of the strained members are taken as the points 
of application of the stresses, shall the bending stress exceed 
15,000 pounds per sq. in. 

VII. Plate girders shall be proportioned upon the supposi- 
tion that the bending or chord stresses are resisted entirely by 
the upper and lower flanges, and that the shearing or web 
stresses are resisted entirely by the web plates. 

VIII. When the length of beams and girders is more than 
thirty times their width, their flanges in compression shall be 
stayed against transverse crippling, and the unsupported width 
of any plate in compression shall not exceed thirty times its 
thickness. 

IX. Shearing stresses in web plates shall not be greater 
than 4,000 pounds per sq. in., and no web or similar plate shall 
be less than -{-^ inch in thickness. When the least thickness of 
the web is less than gV the depth of a girder, the web shall be 
stiffened at intervals not over twice the depth of the girder. 

X. All rivets and bolts connecting parts of girders or 
trusses must be so spaced that the shearing stress shall not 
exceed 7,500 pounds per sq. in. 

XI. In members subject to tensile stress, full allowance shall 
be made for reduction of section by rivet holes or otherwise. 

XII. Any member subjected to a bending stress from local 
loads, in addition to the stress produced by its position as a 
member of the structure, must be proportioned to resist the 
combined stresses. 



Art. 88.] FRANKLIN SQUARE BRIDGE, 697 

13. The stresses allowed in members made of steel, gener- 
ally, will be one-half larger than those specified for wrought 
iron ; . . . 

Eye-Bars and Pins. 

16. Bars which are to be placed side by side in the struct- 
ure shall be bored at the same temperature, and of such equal 
lengths that when the bars are piled on each other the pins 
may pass through all the holes at both ends without driving. 

17. Pin holes shall not be bored more than -^^ inch larger 
than the diameter of the corresponding pins. . . 

18. Any loop attached to a pin must fit it perfectly through- 
out its semi-circumference. 



Compression Members — Riveted Work. 

23. The open sides of all trough-shaped sections shall, at 
distances not exceeding the width of the member, be stayed 
diagonally by lattice bars in size duly proportioned to such 
width. 

24. Whenever it is necessary to reduce the pressure upon a 
pin to the limit prescribed, the pin holes shall be reinforced by 
additional plates, which must contain enough rivets to transfer 
the proportion of pressure to the member. 

27. The pitch of rivets in all classes of work shall not ex- 
ceed six inches, nor sixteen times the thickness of the thinnest 
outside plate, nor be less than three diameters of the rivet ; 
the rivets used shall be generally ^ and ^ inch in diameter. 
In compression members the pitch of rivets for a space from 
its end of twice the breadth or width of a member, shall not 
be over four times the diameter of the rivets. 



698 SPECIFICATIONS. [Art. 88. 

Bed and Buckled Plates. 

38. All bed plates shall be of such dimensions that the 
greatest pressure upon the masonry shall not exceed 250 
pounds per sq. in. 

40. The buckled plates for the roadways shall not be less 
than y^6 ir^ch thick. 



Tests of Material. 

47. All wrought iron must be tough, fibrous and uniform in 
character, and shall have a limit of elasticity of not less than 
26,000 pounds per sq. in. .... . 

• ••••••• 

52. The steel used must be of a uniform and suitable qual- 
ity, known as mild steel. It must have an ultimate tensile 
strength of 70,000 pounds per sq. in. of full section, and when 
marked off in foot lengths, have an ultimate stretch of 10 per 
cent, in the total length, and of at least 15 per cent, in the foot 
which includes the fractured section, showing a reduction in 
section at the point of fracture of at least 25 per cent. It must 
have an elastic limit of 40,000 pounds per sq. in., and a modu- 
lus of elasticity between 26,000,000 and 30,000,000 pounds. 
Small specimens, one foot long and of uniform section of one 
sq. in., cut without work from finished shapes, shall have an 
ultimate resistance of 75,000 pounds and stretch 15 per cent. 
..••• *•■ 

55. Castings must be smooth, free from air holes, cinders, 
and other imperfections, and of good, tough cast iron ; gener- 
ally they shall not be less than ^ inch thick. 



Art. 90.] RAIL WA V SPECIFICA TIONS. 699 



Art. 89. — General Specifications. 

Some general specifications in use at the present time by 
one of the large railroads of the country, require that the 
working stresses per square inch shall not exceed the follow- 
ing values : 

LBS. 

Wrought iron in tension 10,000.00 

Wrought iron in compression 8,000.00 

Wrought iron rivets in shear 6,500.00 

Shearing in web plate of pla-te girders 5,000.00 

Shearing in pins 6,000.00 

Timber in tension 400.00 to 500.00 

Timber in compression 200.00 to 300.00 



Art. 90. — New York, Chicago & St. Louis Railway Specifications. 

" Tensile Members. — ....... 

Where the floor beams are suspended the loops shall be 
double at each panel joint, and the strain . . . must not 
exceed 4,000.00 pounds per sq. in. . 

Compression Members — . . ..... 

The thickness of metal in columns must not be less than 
3V of the width of plates between supports, nor less than y^ 
inch when both faces are accessible for painting, and -^-^ inch 
when one face only is accessible. ... 

When lattice work is used, the distance between rivets 
must not be less than length of segment of equal strength per 
sq. in., as the column itself and the lattice bars must be calcu- 
lated as struts resisting the difference in the strengths per sq. 
in. of the column, and that of its weakest segment acting 
singly without lateral supports. 

In I-beams the compression per sq. in. in the compressed 
flanges must not exceed 



700 SPECIFIC A TIONS. [Art. 90. 

I 40,000 



5i+ ^ 



5,000/^^ 



where / = length of beam in inches and b = breadth in inches 
of the compressed flange. The shearing stress per sq. in. in 
web of I-beam must not exceed 



I 40,000 

^ I + 



3,000/' 



in which d = distance in inches between flanges or stiffeners, 
measured on a line inclined at 45°, and t = thickness of web 
in inches. 

Connections and Attachments. — ..... 

Tensile stresses will not be allowed in a transverse direction 
to the fibres of the iron. Shearing stresses will not be allowed 
in a direction parallel to the fibres of the iron. 



Pins and Rivets. — ..... 
. . . the mean pressure on semi-intrados of pin and rivet 
holes must not be more than 12,500.00 pounds per sq. in. . . 



For Bending, the maximum allowed on the outside fibres of 
timber shall be 1,000.00 pounds per sq. in. 

Bed Plates and Friction Rollers. .... 
. . . the friction rollers must be so proportioned that the 

pressure per lineal inch of roller does not exceed \/540,ooo X d^ 

in which d represents the diameter of roller in inches." 

h 



Art. 91.] PLATTSMOUTH BRIDGE. 7OI 



Art. 91. — Plattsmouth Bridge. 

The following information in regard to the combined steel 
and iron bridge at Plattsmouth, Neb. (George S. Morrison, 
Chief Engineer), is taken from the "Railroad Gazette" of 17th 
Dec., 1880. This bridge carries a single track for the C. B. & 
Q. R.R. 

" The top chords and inclined end posts are riveted steel 
members formed of plates and angles, measuring 28 ins. wide 
by 19 ins. deep over all, the under side being open and laced. 
In the manufacture of these pieces the steel was first punched 
with ^-inch holes, then assembled and the hole? reamed to I 
inch, and then riveted without taking apart, the rivets being of 
low carbon steel. The maximum compressive strain allowed 
to these members is 15,000.00 pounds per sq. in., the sections 
being so proportioned as to carry this strain on the two side 
pieces of the member, the central part of the top plate being 
relied upon only for lateral stiffness. The connection between 
the top chord and end posts, and between the end posts and 
bolsters are pin connections, all parts being entirely of steel. 
On these pins the pressure per square inch, measured on the 
diameter and not on the semi-intrados, is limited to 20,000 
pounds per sq. in. 

The steel bars in the bottom chord and the main ties were 
rolled by the Kloman process in a universal mill, the motion 
being reversed while the bar is still between the rolls, the heads 
being subsequently forged into shape with a steam hammer, 
and the whole bar afterwards annealed. Of seven full-sized 
bars which were tested to breaking, not one broke or showed 
any weakness in or near the head. . . The maximum strain 
allowed on steel in tension is 15,000 pounds per sq. in., this 
occurring only in the middle panels of the bottom chord, and 
being reduced to 12,500 in the end panels ; in the web the 
strain per square inch varies from 10,000 pounds at the centre 



702 SPECIFICATIONS. [Art. 9 1. 

to about 12,500 in the end ties, except under the extraordinary 
supposition of the entire weight on the driving wheels of two 
75-ton locomotives being carried entirely by the same system ; 
in this case the maximum strain on the end diagonals will 
slightly exceed 14,000 pounds per sq. in. 

The counter ties and lateral rods are also of steel. . . The 
strain on the counters is always less than 10,000 pounds per 
sq. in. ; that on the laterals is limited to 22,000 pounds. Tests 
made of these light steel bars showed a superior proportional 
excellence fully equal to that commonly found in small sec- 
tions of wrought iron as compared with large sections. 

The intermediate posts are of wrought iron, each post con- 
sisting of two channels laced at the sides. 

All the pins, except the lateral ones, are of 4||- inches diam- 
eter. 

Steel. 

The steel used in the Plattsmouth bridge was manufactured 
by Hussey, Howe & Co., of Pittsburgh, in an open hearth fur- 
nace. The specifications required that a sample (about 5^ inch 
diameter) should be taken from every melt, and that this bar 
should bend cold 180 degrees around its own diameter without 
cracking ; that it should have an elastic limit of at least 50,000 
pounds, and an ultimate strength of at least 80,000 pounds ; 
and that it should elongate 12 per cent, before breaking and 
show a reduction of 20 per cent, at the point of fracture. The 
percentage of carbon was fixed at 0.35. 

A difference in the strength of large and small sized bars 
corresponding to that which exists in iron bars was found in 
the steel. The finished bars measured 6x1}^ inches to i^^ 
inches ; when tested in the Government machine at Water- 
town, Mass., they were found to have an elastic limit of 37,000 
pounds, and ultimate resistances of 66,000 to 73,000 pounds 
per square inch. The modulus of elasticity below the elastic 



Art. 92.] STEEL CABLE WIRE. 703 

limit was exceedingly uniform. Smaller sizes, used in counters 
and laterals, approximated closely in their strength and elastic 
limit to the test sam,ples. 



Art. 92.— Specifications for Steel Cable Wire for the East River 

Suspension Bridge. 



3. The general character of the wire is as follows : it must 
be made of steel; it must be hardened and tempered ; and, 
lastly, it must all be galvanized. 

4. The size of the wire shall be No. 8 full, Birmingham 
gauge. ....... 

5. Each wire must have a breaking strength of no less than 
3,400 pounds. This corresponds in wire weighing 14 feet to 
the pound, to a rate of 160,000 pounds per square inch of solid 

section. The elastic limit must be no less than ^- of the 

100 

breaking strength, or, 1,600 pounds. Within this limit of elas- 
ticity, it must stretch at a uniform rate corresponding to a 
modulus of elasticity of not less than 27,000,000 nor exceed 
29,000,000. ....... 

Mode of Testing, 



There will be four kinds of tests. 

Firstly. — One ring in every forty (40) will be tested as fol- 
lows : a piece of wire sixty (60) feet long, will be cut off from 
either end of the ring, and it will then be placed in a vertical 
testing machine. An initial strain of 400 pounds is now ap- 
plied, which should take out every crook and bend. A vernier 

gauge, capable of being read to of one foot, is so at- 



704 SPECIFICATIONS] [Art. 92. 

tached as to indicate the stretch of 50 feet of the wire. Suc- 
cessive increments of 400 pounds strain are then applied, and 
the vernier read each time, until a strain of 1,600 pounds is 
reached. 

The conditions now are as follows: that the amount of 
stretch for each of these increments shall be the same, and that 
the total stretch between the initial and terminal strains shall 

not be less than ^ of one foot, equal to — — — of the 50 
1,000 100,000 

feet. And furthermore, on reducing the strain to 1,200 

pounds there shall be a permanent elongation not exceeding 

■ of its lenp;th. 

100,000 

The same wire will then be subjected to a breaking strain, 

and the total amount of stretch noted. The minimum strength 

required is 3,400 pounds, equal to an ultimate strength of 

160,000 pounds per square inch. The minimum stretch, when 

broken, shall have been 2 per cent, in 50 feet, and the diameter 

of the wire at the point of fracture shall not exceed — — of one 
^ 100 

inch. 

• ••••••• 

Fourthly. — Every ring will be subjected to a bending test 
by cutting off from each ring a piece of wire one foot long, and 
coiling it closely and continuously around a rod one half inch 
in diameter, when, if it breaks it will be rejected. 



Straight Wire, 

% 
9. All the wire . . . must be " straight " wire ; that is 

to say, when a ring is unrolled upon the floor the wire behind 

must lie perfectly straight and neutral, without any tendency 

to spring back in the coiled form, as is usually the case. This 



Art. 94.] STEEL WIRE ROPES. 705 

straight condition must not be produced by the use of straight- 
ening machines of any kind, as they only injure the strength 
and elasticity of the wire. . . . . ." 



Art. 93. — Specifications for Steel Wire Ropes for the Over-Floor Stays 
and Storm Cables of the East River Suspension Bridge. 



3. The steel from which the wire for these ropes is made 
must be of a uniform and suitable quality, and after drawn 
must be thoroughly and evenly galvanized throughout. 

4. The galvanized wire must have an ultimate strength of 
150,000 pounds per square inch of full section. When tested 
in lengths of five feet it must stretch no less than three and 
one-half per cent, of its length, and in lengths of one foot it 
must stretch no less than four per cent. 

5. It must be capable of being bent continuously around a 
rod of three times the diameter of the wire, without fracture. 

6. The modulus of elasticity must not vary more than 
2,000,000 pounds, nor exceed 30,000,000 pounds. 

7. It must have a limit of elasticity of not less than 70,000 
pounds per square inch. 



Art. 94. — Specifications for Steel Suspenders, Connecting Rods, Stirrups 
and Pins, for the East River Suspension Bridge. 



All of th& steel used must be of a uniform and suitable 
quality, known as " Mild Steel." It must have an ultimate 
tensile strength of 75,000 pounds per square inch of full sec- 
tion, and an ultimate stretch of no less than 15 per cent, in one 
foot of length, including the fractured section ; and a reduc- 
tion of no less than 25 per cent, of area at the point of fracture. 
It must have an elastic limit of no less than 45,000 pounds per 
square inch, and a modulus of elasticity between 26,000,000 
45 



70^ SPECIFICA TIONS. [Art. 95. 

and 30,000,000 pounds per square inch. Specimens turned 
down from full-sized rods to an area of one square inch, or less, 
must show a greater strength per square inch, and a greater 
elongation than that called for in the full section." 



Art. 95. — Specifications for Certain Steel Work . . . East River 

Bridge, 1881. 



All of the steel used in this work must be of a mild, uni- 
form, elastic and ductile quality, suitable for bridge members. 
Siemens-Martin or open-hearth steel, or Bessemer steel under 
the Hay process, will be preferred. 

Specimens of the steel proposed to be used must be fur- 
nished by each bidder. Two specimens, direct from the rolls, 
each I inch square and 24 inches long, are required. 

• •••• ••• 

All of the steel must be capable of sustaining a tensile 
strain in every full-sized round or flat bar of not less than 
70,000,000 pounds per square inch of cross section. It must 
have an elastic limit in all shapes of no less than 40,000 pounds 
per square inch. A modulus of elasticity of not less than 
26,000,000 nor more than 30,000,000 pounds per square inch. 

An ultimate elongation of 10 per cent, of the full length of 
uniform sections, and 15 per cent, in one foot of length, inclu- 
sive of fractured section, is also required. The area of the 
reduced section at the point of fracture must not exceed 80 
per cent, of the original section. * 

Small specimens of one foot in length, of even section of 
one square inch, or less, should rieach in tensile strength 75,000 
pounds per square inch, with a modulus and limit of elasticity, 
and reduction of area before mentioned, and an ultimate 
stretch of 15 per cent. 

All round or flat bars, or flat pieces cut from the web of any 
shaped bars, must be capable of being bent cold for 180° to a 



Art. 95.] EAST RIVER STEEL WORK. JOJ 

curve whose diameter is no greater than the thickness of the 
bar, and that without cracking. 

The rivets must be made of very ductile steel particularly 
adapted for that use. 

The rods from which the rivets are made must, when tested, 
have a tensile strength of not less than 70,000 pounds per 
square inch, and elongate at least 20 per cent, in a length of 
one foot, and shall reduce at the point of fracture 30 per cent. 

If the minimum is reached in any one of these requirements 
the others must be exceeded by at least 10 per cent. The rod 
must be capable of being bent cold under a hammer 180°, and 
the inner surfaces brought in contact without producing any 
fracture. 



Cold straightening must be avoided, and when resorted to, 
the piece so straightened must be annealed afterwards, and all 
pieces, of which any portion for any cause is reheated, the 
whole must be annealed and very slowly cooled ; and all pieces 
in which, from test or otherwise, a want of uniformity is sus- 
pected, must be annealed if required by the engineer. 

All rivet holes must be drilled, unless some system of 
punching and reaming approved by the engineer be followed, 
whereby all of the compressed section around the punched 
hole be cut away. 

The spacing must be accurately done, as no gauging or 
drifting will be allowed." 



CHAPTER XIII. 

The Fatigue of Metals. 

Art. 96. — Woehler's Law. * 

In all the preceding pages, that force or stress, which, by a 
single or gradual application, will cause the failure or rupture 
of a piece of material, has been called its "ultimate resistance." 
It has long been known, however, that a stress less than the 
ultimate resistance may cause rupture if its application be re- 
peated (without shock) a sufficient number of times. Preced- 
ing 1859 ^o experiments had been made for the purpose of 
establishing any law connecting the number of applications 
with the stress requisite for rupture, or, with the variation 
between the greatest and least values of the applied stress. 

During the interval between 1859 ^.nd 1870, A. Wohler, 
under the auspices of the Prussian Government, undertook the 
execution of some experiments, at the completion of which he 
had established the following law : 

Rupture may be caused not only by a force which exceeds the 
ultimate resistance, but by the repeated action of forces alternately 
rising and falling between certain limits, the greater of which is 
less than the ultimate resistance ; the number of repetitions re- 
quisite for rupture being an inverse function both of this vari- 
ation of the applied force and its upper limit. 

This phenomenon of the decrease in value of the breaking 
load with an Increase of repetitions, is known as ^'' the fatigue 
of materia Is.'* 

Although the experimental work requisite to give Wohler's 



Art. 9;.] 



EXPERIMENTAL RESULTS. 



709 



law complete quantitative expression in the various conditions 
of engineering constructions can scarcely be considered more 
than begun, yet enough has been done by Wohler and Span- 
genberg to establish the fact of metallic fatigue, and a few 
simple formulae, provisional though they may be. The im- 
portance of the subject in its relation to the durability of all 
iron and steel structures is of such a high character that a 
synopsis of some of the experimental results of Wohler and 
Spangenberg will be given in the next Article. , 

Art. 97. — Experimental Results. 

The experiments of Wohler are given in " Zeltschrift fiir 
Bauwesen," Vols. X., XIII., XVI. and XX., and those of Span- 
genberg may be consulted in " Fatigue of Metals," translated 
frorti the German of Prof. Ludwig Spangenberg, 1876. 

These results show in a very marked manner the effect of 
repeated vibrations on the intensity of stress required to pro- 
duce rupture. 

Spangenberg states that ''the experiments show that vibra- 
tions may take place between the following limits with equal 
security against rupture by tearing or crushing : 



~|- 17,600 and — 17,600 lbs. per sq. in 
Wrought iron -^ -(- 33,000 and o 

-|- 48,400 and -f- 26,400 

+ 30,800 and — 30,800 
Axle cast steel -^ -|- 52,800 and o 

-f- 88,000 and 4" 38,500 

r4" 55.000 and o 

J + 77,000 and + 27,500 

I -|- 88,000 and + 44,000 

I -\- 99,000 and -f- 66,000 



Spring steel not hardened. 



And for axle cast steel in shearing : 



+ 24,200 and — 24,200 lbs. per sq. in. 
4" 41,800 and o 



(( «{ (i 



7IO 



FATIGUE OF METALS, 



[Art. 97. 



Phoe7iix Iron in Tension. 



POUNDS STRESS PER 


NUMBER 


POUNDS STRESS PER 


NUMBER 


SQUARE INCH. 


OF REPETITIONS. 


SQUARE INCH. 


OF REPETITIONS. 


From to 52,800 


800 rupture. 


From to 39,600 


480,852 rupture. 


From to 48,400 


106,910 rupture. 


From to 35,200 


10,141,645 rupture. 


From to 44,000 


340,853 rupture. 


From 22,000 to 48,400 


2,373,424 rupture. 


From to 39,600 


409,481 rupture. 


From 26,400 to 48,400 


4,000,000 not broken. 



Westphalia Iron in Tension. 



From to 52,800 


4,700 rupture. 


From to 39,600 


180,800 rupture. 


From to 48,400 


83,199 rupture. 


From to 39,600 


596,089 rupture. 


From to 48,400 


33,230 rupture. 


From to 39,600 


433,572 rupture. 


From to 44,000 


136,700 rupture. 


From to 35,300 


280,121 rupture. 


From to 44,000 


159,639 rupture. 


From to 35,200 


566,344 rupture. 



Firth &= Sons'* Steel in Tension. 



From to 66,000 


83,319 rupture. 


From to 55,000 


103,540 rupture. 


From to 60,500 


168,396 rupture. 


From to 53,900 


12,200,000 not broken. 


From to 55,000 


133,910 rupture. 


From to 53.^00 


229,230 rupture. 


From to 55,000 


185,680 rupture. 


From to 52,800 


692,543 rupture. 


From to 55,000 


360,235 rupture. 


From to 52,800 


12,200,000 not broken. 


From to 55,000 


186,005 rupture. 


From to 50,600 







Krupp''s Axle Steel ijt Tetisioii. 



From to 88,000 


18,741 rupture. 


From to 55,000 


473,766 rupture. 


From to jj^ooo 


46,286 rupture. 


From to 52,800 


13,600,000 not broken. 


From to 66,000 


170,000 rupture. 


From to 50,600 


12,200,000 not broken. 


From to 60,500 


123,770 rupture. 











Art. 97.] 



EXPERIMENTAL RESULTS. 



711 



Phosphor Bronze {u7i'workea) in Tension. 



POUNDS STRESS PER 
SQUARE INCH. 


NUMBER 
OF REPETITIONS. 


POUNDS STRESS PER 
SQUARE INCH. 


NUMBER 
OF REPETITIONS. 


From to 27,500 
From to 22,000 
From to 16,500 


147,850 rupture. 

408,350 rupture. 

, 2,731,161 rupture. 


From to 13,750 
From to 13,750 


1,548,920 rupture. 
2,340,000 rupture. 







Phosphor Bronze (wroughf) in Tensiojt. 



From o to 22,000 
From o to 16,500 



53,900 rupture. 
2,600,000 not broken. 



From o to 13,750 



1,621,300 rupture. 



Comino7i Bronze in Teiision. 



From o to 22,000 
From o to 16,500 



4,200 rupture. 
6,300 rupture. 



From o to 11.000 



5,447,600 rupture. 



Phcejtix Iron in Flexure {one direction only). 



From to 60,500 


169,750 rupture. 


From to 39,600 


4,035,400 ruptnre. 


From to 55,000 


420,000 rupture. 


From to 35,200 


3,420,000 rupture. 


From to 49,500 


481,975 rupture. 


From to 33,000 


4,820,000 not broken. 


From to 44,000 


1,320,000 rupture. 











Westphalia Iron iji Flexure {one direction only). 



From o to 52,250 
From o to 49,500 
From o to 46,750 



612,065 rupture. 
457,229 rupture. 
799,543 rupture. 



From o to 44,000 
From o to 39,600 



1,493,511 rupture. 
3.5871509 rupture. 



712 



FATIGUE OF METALS. 



[Art. 97. 



Homogeneous Iron m Flexure {one direction only). 



POUNDS STRESS PER 


NUMBER 


POUNDS STRESS PER 


NUMBER 


SQUARE INCH. 


OF REPETITIONS. 


SQUARE INCH. 


OF REPETITIONS. 


From to 60,500 


169,750 rupture. 


From to 39,600 


4^035 i4oo rupture. 


From to 55,000 


420,000 rupture. 


From to 35,200 


3,420,000 not broken. 


From to 49,500 


481,975 rupture. 


From to 33,000 


48,200,000 not broken. 


From to 44,000 


1,320,000 rupture. 











Firth 6r» Sons' Steel in Flexure {one direction only). 



From to 63,250 


281,856 rupture. 


From to 52,250 


578,323 rupture. 


From to 60,500 


266,556 rupture. 


From to 49,500 


5,640,596* rupture. 


From to 55,000 


1,479,908 rupture. 


From to 49,500 


13.700,000 not broken. 



* Accide'ntal. 
Krupp's Axle Steel in Flexure {one direction only). 



From to 77,000 


104,300 rupture. 


From to 55,000 


729,400 rupture. 


From to 66,000 


317,275 rupture. 


From to 55,000 


1,499,600 rupture. 


From to 60,500 


612,500 rupture. 


From to 49,500 


43,000,000 not broken. 



Krupp's Spring Steel in Flexure {one direction only). 



From to 


110,000 


39,950 rupture. 


From 72,600 to 


110,000 


19,673.300 not broken. 


From to 


88,000 


117,000 rupture. 


From 66,000 to 


09,000 


33,600,000 not broken. 


From to 


66,000 


468,200 rupture. 


From 44,000 to 


88,000 


35,800,000 not broken. 


From to 


55.000 


40,600,000 not broken. 


From 44.000 to 


88,000 


38,000,000 not broken. 


From to 


49.500 


32,942,000 not broken. 


From 61,600 to 


88,000 


36,000,000 not broken. 


From 88,000 to 


132,000 


35,600,000 not broken. 


From 27,500 to 


77,000 


36,600,000 not broken. 


From 99,000 to 


132,000 


33,478,700 not broken. 


From 33,000 to 


77,000 


31,152,000 not broken. 



Art. 97.] 



EXPERIMENTAL RESULTS. 



713 



Phosphor Bronze in Flexure {one direction only). 



POUNDS STRESS PER 
SQUARE INCH. 


NUMBER 
OF REPETITIONS. 


FOUNDS STRESS PER 
SQUARE INCH. 


NUMBER 
OF REPETITIONS. 


From to 22,000 
From to 19,800 


862,980 rupture. 
8,151,811 rupture, 


From to 16,500 
From to 13,200 


5,075,169 rupture. 
10,000,000 not broken. 



Common Bronze in Flexure {one direction only). 



From o to 22,000 
From o to 19,800 



102,659 rupture. 
151,310 rupture. 



From o to 16,500 
From o to 13,200 



8371760 rupture. 
10,400,000 not broken. 



Phoenix Iron in Torsion {both directions). 



- 35,200 to + 35,200 


56,430 rupture. 


- 24,200 to + 24,200 


3,632,588 rupture. 


- 33,000 to + 33,000 


99,000 rupture. 


- 22,000 to + 22,000 


4,917,992 rupture. 


- 28,600 to + 28,600 


479,490 rupture. 


- 19,800 to + 19,800 


19,186,791 rupture. 


- 26,400 to + 26,400 


909,810 rupture. 


- 17,600 to + 17,600 


132,250,000 not broken. 



English spindle Iron in Torsion {both directions). 



- 37.400 


to 


+ 37.400 


204,400 rupture. 


- 30,800 


to + 30,800 


979,100 rupture. 


- 37.400 


to 


+ 37.400 


147,800 rupture. 


- 28,600 


to + 28,600 


1,142,600 rupture. 


- 3S.200 


to 


+ 35,200 


911,100 rupture. 


- 28,600 


to + 28,600 


595,910 rupture. 


- 35.200 


to 


+ 35.200 


402,900 rupture. 


- 26,400 


to + 26,400 


3,823,200 rupture. 


- 33.000 


to 


+ 33,000 


1,064,700 rupture. 


- 26,400 


to + 26,400 


6,100,000 not broken. 


- 33,000 


to 


+ 33.000 


384,800 rupture. 


- 22,000 


to + 22,000 


8,800,000 not broken. 


- 30,800 


to 


+ 30,800 


1,337,700 rupture. 


- 22.000 


to + 22,000 


4,000,000 not broken. 



714 



FATIGUE OF METALS, 



[Art. 97. 



Krupp's Axle Steel in Torsion {both direciio7ts). 



POUNDS STRESS PER 


NUMBER. 


POUNDS STRESS PER 


NUMBER 


SQUARE 


INCH. 


OF REPETITIONS. 


SQUARE 


INCH. 


OF REPETITIONS. 


- 44,000 to 


+ 44,000 


367,400 rupture. 


- 46,200 to 


+ 46,200 


55,100 rupture. 


- 39,600 to 


+ 39,600 


925,800 rupture. 


- 37,400 to 


+ 37,200 


797,525 rupture. 


- 37,400 to 


+ 37i40o 


4,900,000 not broken. 


- 35,200 to 


+ 35»2oo 


1,665,580 rupture. 


- 35,200 to 


+ 35,200 


4,800,000 not broken. 


- 33,000 to 


+ 33,000 


4,163,37s rupture. 


- 33,000 to 


+ 33,000 


5,000,000 not broken. 


- 33,000 to 


+ 33,000 


45,050,640 rupture. 



In Art. 33 will be found some experiments by the late Capt. 
Rodman, U. S. A., on the fatigue of cast iron, but they are 
sufficient in number and character to show the general effect 
only, and give no quantitative results. 

The specimens used in all the preceding experiments were 
small. 

During i860, '61 and '62, Sir Wm. Fairbairn constructed a 
built beam of plates and angles with a depth of 16 inches, clear 
span of 20 feet, and estimated centre breaking load of 26,880 
pounds. 

This beam was subjected to the action of a centre load of 
6,643 pounds, alternately applied and relieved eight times per 
minute ; 596,790 continuous applications produced no visible 
alterations. 

The load was then increased from one-fourth to two- 
sevenths the breaking weight, and 403,210 more applications 
were made without apparent injury. 

The load was next increased to two-fifths the breaking 
weight, or to 10,486 pounds; 5,175 changes then broke the 
beam in the tension flange near the centre. 

The total number of applications was thus 1,005,175. 

The beam was then repaired and loaded with 10,500 pounds 
at centre 158 times ; then with 8,025 pounds 25,900 times, and 



Art. 98.] LAUNHARDT'S FORMULM. 715 

finally with 6,643 pounds enough times to make a total of 
3,150,000. 

In these experiments the load was completely removed 
each time. 

It is thus seen that vibrations (without shock) with one- 
fourth the calculated breaking centre load produced no appar- 
ent effect on the resistance of the beam, but that two-fifths of 
that load caused failure after a comparatively small number of 
repetitions. 

It is probable that the breaking centre load was calculated 
too high, in which case the ratios \ and f should be somewhat 
increased. 



Art. 98. — Formulae of Launhardt and Weyrauch. 

Let R represent the intensity (stress per square unit of sec- 
tion) of ultimate resistance for any material in tension, com- 
pression, shearing, torsion or bending ; R will cause rupture at 
a single, gradual application. But the material may also be 
ruptured if it is subjected a sufificient number of times, and 
alternately, to the intensities Pand Q, Q being less than /* and 
both less than R^ while all arc of the same kind. When Q = o 
let P = W, and let D = P — Q. W is called the " primitive 
safe resistance," since the bar returns to its primitive unstressed 
condition at each application. In the general case i^ is called 
the *' working ultimate resistance." 

By the notation adopted : 

P=Q + -D (I) 

But by Wohler's law, /* is a function of D ; or, 

P = f{D) (2) 

A sufficient number of experiments have not yet been made 



71^ FATIGUE OF METALS. [Art. 98. 

in order to completely determine the form of the function 

It is known, however, that : 

For = o ; P = D = W; 
and for 

D = o; P=z Q = R, 

Provisionally, Launhardt satisfies these two extreme con- 
ditions by taking : 

P^^^n-:^^iP-Q) ... (3) 



Even at these limits this is not thoroughly satisfactory, for 

o 
By solving Eq. (3) : 



when D = o, P = — {R — W), or, indeterminate. 



But if the least value of the total stress to which any mem- 
ber of a structure is subjected is represented by inin B, and its 

ijttfi B O 

greatest value by max B, there will result = = ~ . 

max B P 

Hence: 

V ^ W max ^; • • • • V5; 

which is Launhardt's formula. In the preceding Article some 
values of W are shown. In applying Eq. (5) it is only neces- 
sary to take the primitive safe resistance, W^ for the total 
number of times which the structure will be subjected to loads. 



Art. 98.] WEYRAUCH'S FORMUL/E. 7^7 

Since bridges are expected to possess an indefinite duration of 
life, in such structures that number should be indefinitely 
large. 

Eq, (5), it is to be borne in mind, is to be applied when the 
piece is always subjected to stress of one kind^ or in one direction 
only. It agrees well with some experiments by Wohler on 
Krupp's untempered cast spring steel. 

If the stress in any piece varies from one kind to another, 
as from tension to compression, or vice versa ; or from one 
direction to another, as in torsion on each side of a state of no 
stress, Weyrauch has established the following formula by a 
course of reasoning similar to that used by Launhardt. 

If the opposite stresses, which will cause rupture by a cer- 
tain number of applications, are equal in intensity, and if that 
intensity is represented by 5, then will 5 be called the " vibra- 
tion resistance"; this was established by Wohler for some 
cases, and some of its values are given in the preceding 
Article. 

Let -f- P and — P' represent two intensities of opposite 
kinds or in opposite directions, of which P is nmnerically the 
greater. Then \i D = P -\- P' \ 

P= D - P\ 
The two following limiting conditions will hold : 

¥ov P' = o\ P=. B = W: 
ForP' = S; P= S = J^Z>. 

But by Wohler's law, P = / {-D), and the two limiting con- 
ditions just given will be found to be satisfied by the pro- 
visional formula : 



71 8 FATIGUE OF METALS. [Art. 98. 

By the solution of Eq. (6) : 

W- S P'' 



P 



„,f W- S P'\ , . 



If, without regard to kind or direction, max B is numerically 

the greatest total stress which the piece has to carry, while 

max B' is the greatest total stress of the other kind or direc- 

P ifiax B' 

tion, then will -=r- = ^ . Hence, there will result the fol- 

P max B 

lowing, which is the formula of Weyrauch : 

V W max B) ' ' ' ' ^^^ 

Eqs. (5) and (8) give values of the intensity P which are to 
be used in determining the cross section of pieces designed to 
carry given amounts of stress. If ;/ is the safety factor and F 
the total stress to be carried, the area of section desired will be : 

A -''^• 

P . 

in which — is the greatest working stress permitted. 

If for wrought iron In tension W — 30,000 and R — 50,000, 
Eq. (5) gives: 

7-, / , 2 min B 

P = 30,000 ( I + B 

\ 3 max B^ 

Hence, if the total stress due to fixed and moving loads in 
the web member of a truss is max B = 80,000 pounds, while 
that due to the fixed load alone is min B = 40,000, there will 
result : 

T^ f , 2 4o,ooo\ 

p = 30,000 ( I + - • w~~) = 40,000. .» 

\ 3 80,000/ 



Art. 99.] INFLUENCE OF TIME. 719 

In such a case the greatest permissible working stress with 
a safety factor of 3 would be about 13,300 pounds. 

For steel in tension, i( W = 50,000 and R = 75,000: 



P = 50,000 ^i + 



1 7m 71 B 

2 max B 



For wrought iron in torsion, if 5 = 18,000 and W — 24,000, 
Eq. (8) will give : 

I max B'^ 



„ /I max B \ 

P = 24,000 I ^ . 

\ 4 7nax B J 



Other methods based on Wohler's experiments have been 
deduced by Miiller, Gerber and Schaffer, of which synopses 
may be found in Du Bois' translation of Weyrauch's *' Struct- 
ures of Iron and Steel." 



Art. 99. — Influence of Time on Strains. 

In the section '' elevation of ultimate resistance a7td elastic 
limit,'' in Art. 32, the effect of prolonged tensile stress and 
subsequent rest between the elastic limit and ultimate resist- 
ance, was shown to be the elevation of both those quantities. 
It is a matter of common observation, however, that if a piece 
of wrought iron be subjected to a tensile stress nearly equal to 
its ultimate resistance, and held in that condition, that the 
stretch will increase as the time elapses. 

Experiments are still lacking which may show that a piece 
of metal can be ruptured by a tensile stress much below its 
ultimate resistance. It may be indirectly inferred, however, 
from experiments on flexure, that such failure may be pro- 
duced, as the following by Prof. Thurston will show. 

A bar 10 parts tin and 90 parts copper, i X i x 22 inches 
and supported at each end, sustained about 65 per cent, of its 



720 FATIGUE OF METALS. [Art. 99. 

breaking load at the centre for five minutes. During that time 
its deflection increased 0.021 inch. The same bar sustained 
1,485 pounds at centre for 13 minutes and then failed. 

A second bar of the same size, but 90 parts tin and 10 parts 
copper, was loaded at the centre with 160 pounds, causing a 
deflection of 1.294 inches. After 10 minutes the deflection 
had increased 0.025 inch; after one day, i. 00 inch; after two 
days, 2.00 inches ; and after three days, 3.00 inches, when the 
bar failed under the load of 160 pounds. 

Another bar of the same size showed remarkable results ; it 
was composed of 90 parts zinc and 10 parts copper. It gave 
the same general increase of deflection with time, but eventually 
broke under a centre load which ran down from 1,233 to 911 
pounds, after holding the latter about three minutes. 

A bar of the same size and 96 parts copper with 4 parts tin, 
after it had carried 700 pounds at centre for sixty minutes was 
loaded with 1,000 pounds, with the following results : 

AFTER DEFLECTION. 

o minute 3- 118 inches. 

5 minutes 3-540 " 

1 5 minutes 3 . 660 * * 

45 minutes 4. 102 ** 

75 minutes 7-634 " 

Broke under 1,000 pounds. 

A wrought-iron bar of the same size gave, under a centre 
load of 1,600 pounds : 

AFTER DEFLECTION. 

O minute o. 489 inch. 

3 minutes o . 632 * ' 

6 minutes 0.650 " 

16 minutes 0.660 " 

344 minutes o . 660 * ' 

It subsequently carried 2,589 pounds with a deflection of 
4.67 inches. 



Art. 99.] INFLUENCE OF TIME. 721 

During 1875 and 1876 Prof. Thurston made a number of 
other similar experiments with the same general results. 

Metals like tin and many of its alloys showed an increasing 
rate of deflection and final failure, far below the so-called 
** ultimate resistance." The wrought-iron bars, however, showed 
a decreasing increment of deflection, which finally became zero, 
leaving the deflection constant. 

Whether there may be a point for every metal, beyond 
which, with a given load, the increment of deflection may 
retain its value or go on increasing until failure, and below 
which this increment decreases as the time elapses, and finally 
becomes zero, is yet undetermined, but seems probable. 

It does not follow, therefore, that the principle enunciated 
in the section named at the beginning of this Article, is to be 
taken without qualification. If " rest " under stress, too near 
the ultimate resistance, be sufficiently prolonged, it has been 
seen that it is possible that failure may take place. 

In verifying some experimental results by Herman Haupt, 
determined over forty years ago. Prof. Thurston tested three 
seasoned pine beams about i^ inches square and 40 inches 
length of span, and found that 60 per cent, of the ordinary 
** breaking load" caused failure at the end of 8, 12 and 15 
months. In these cases the deflection slowly and steadily in- 
creased during the periods named. 

Two other sets of three pine beams each, broke under 80 
and 95 per cent, of the usual " breaking load," after much 
shorter intervals of time. 

In all these instances it is evident that the molecules under 
the greatest stress " flow " over each other to a greater or less 
extent. In the cases of decreasing increments of strain, the 
new positions afford capacity of increased resistance ; in the 
others, those movements are so great that the distances be- 
tween some of the molecules exceed the reach of molecular 
action, and failure follows. 

In many cases strained portions of material recover partially 
46 



722 FATIGUE OF METALS. . [Art. 99. 

or wholly from permanent set. In such cases a portion of the 
material has been subjected to intensities of stress high enough 
to produce true " flow ** of the molecules, while the remaining 
portion has not. The internal elastic stresses in the latter por- 
tion, after the removal of the external forces, produce in time 
a reverse flow in consequence of the elastic endeavor to resume 
the original shape. 

It is altogether probable that the phenomena of fatigue and 
flow of metals are very intimately associated. Some of the 
prominent characteristics of the latter will be given in the next 
chapter. 



CHAPTER XIV. 

The Flow of Solids. 

Art. 100. — General Statements. 

Although there is no reason to suppose that true solids 
may not retain a definite shape for an indefinite length of time 
if subjected to no external force other than gravity,"^ many 
phenomena resulting both from direct experiment for the pur- 
pose, and incidentally from other experiments involving the 
application of external stress of considerable intensity, show 
that a proper intensity of internal stress (in many cases com- 
paratively low) will cause the molecules of a solid to flow, at 
ordinary temperatures, like those of a liquid. And this flow, 
moreover, is entirely diflerent from, and independent of, the 
elastic properties of the material ; for it arises from a perma- 
nent and considerable relative displacement of the molecules. 
Nor is it to be confounded with that internal " friction " which, 
if an elastic body is subjected to oscillations, causes the ampli- 
tudes to gradually decrease and finally disappear, even in 
vacuo. This latter motion is typically elastic and the retarding 
cause may be considered a kind of elastic friction. 

It is evident that if a mass of material be enclosed on all its 
faces, or outer surfaces, but one or a portion of one, and if 
external pressure be brought to bear on those faces, the mate- 
rial will be forced to move to and through the free surface ; in 

* This, perhaps, may be considered a definition of a true solid. 



724 



FLOW OF METALS. 



[Art. loo. 



other words, the flow of the material will take place in the 
direction of least resistance. 



M 





F G 




__-A 





B — 




ZHZIZM. 





IZIZl 



















D H 


O 


K 


G 



Fig.l 




Fig.2 




The theory of the flow of solids 
to be given is that developed by 
Mons. H. Tresca in his " Memoire 
sur rEcoulement des Corps So- 
lides," 1865. He made a large 
number of experiments on hard 
and soft metals, ceramic pastes, 
sand and shot. 

These different materials all 
manifested the same characteristics 
of flow, which are well shown in 
Fig. 2. ABCDy Fig. i, is supposed 
to be a cylindrical mass of lead 
with circular horizontal section, con- 
fined in a circular cylinder, MNy 
closed at one end with' the excep- 
tion of the orifice O. 

This cylinder is supported on 
the base PNy while the face AB of 
B the lead receives external pressure 
from a close-fitting piston. When 
the pressure is sufficiently increased, 
the face AB in Fig. i sinks to AB 



in Fig. 2, while the column hkHK, 
in the latter figure, is forced to flow 
through the orifice O. 

In Tresca's experiments with 
°D lead, the diameter AB was about 
3.9 inches ; the diameter //"/^ of the 
orifice, from 0.75 in. to 1.5 ins., while the length of the column 
or jet /^iT varied from 0.4 in. to about 24 ins. The total press- 
ure on the face AB varied from 119,000 to 198,000 pounds. 
The initial thickness AD varied from 0.24 inch to 2.4 inches. 



Fig.3 



Art. 1 01.] HYPOTHESES OF TRESCA. 725 

Some experiments exhibiting in a remarkably clear manner 
the flow of metals in cold punching were made by David 
Townsend in 1878, and the results were given by him in the 
" Journal of the Franklin Institute " for March of that year. 
If the dotted rectangle ABFG, Fig, 3, shows the original out- 
line of the middle section of a nut before punching, he found 
that the final outline of the same section would be represented 
by the full lines. The top and bottom faces were depressed 
by the punching, as shown ; the upper width AB remained 
about the same, but the lower, GF^ was increased to CD. Al- 
though the depth of the nut, AC^ was 1.75 inches, the length 
of the core punched out was only 1.063 inches. The density of 
this core was then examined and found to be the same as that 
of the original nut. Hence a portion of the core equal in 
length to 1.75 — 1.063 = 0.687 inch was forced, or flowed, back 
into the body of the nut. Subsequent experiments showed 
that this flow did not take place at the immediate upper sur- 
face AB, nor very much in the lower half of the nut, but that 
it was chiefly confined to a zone equal in depth to about half 
that of the nut, the upper surface of which lies a very short 
distance below the upper face of the nut. The location of this 
zone is shown by the lines ///Tand MN in Fig. 3. 

Tresca's experiments on punching showed essentially the 
same result. 



Art. loi. — Tresca's Hypotheses. 

The central cylinder FGKH, Fig. i of Art. 100 was called 
by Tresca the " primitive central cylinder." As the metal 
flows, this cylinder will be drawn out into the volume of revo- 
lution, Avhose axir; is that of the orifice and whose meridian 
section is FGkKHh, Fig. 2, the diameter FG being gradually 
decreased. 

It was found by experiment that if the original mass AC, 



726 FLOW OF SOLIDS. [Art. lOI. 

Fig. I, was composed of horizontal layers of uniform thickness, 
the reduced mass in Fig. 2 was also composed of the same 
number of layers of uniform thickness, except in the immediate 
vicinity of the central cylinder. 

Tresca then assumed these three hypotheses : 
1°. — The density of the material remains the same whether in 
the cylinder or in the jet ; in other words, the volume of the 
material in the jet and in the cylinder remains constant. 

Let R = radius of the cylinder. 

Let R. — radius of the orifice. 

Let y = variable length of the jet (i, e.y hH\ 

Let D — original depth of material (BC — AD, Fig. i) 

in the cylinder. 
Let d — variable depth of material [BC — AD, Fig. 2) 

in the cylinder. 

Then by the hypothesis just stated : 

R^d = RW - R,y (i) 

2°. The rate of compression along any and all lines parallel 
to the axis of the primitive central cylinder, and taken outside of 
that limit, is constant. 

If, then, the material lying outside of the central cylinder 
be divided into horizontal layers of equal thickness, a very 
small decrease in the variable depth equal to d{a) will cause 
the same amount of material to move or flow from each of 
these layers into the space originally occupied by the central 
cylinder, thus causing a portion of the material previously 
resting over the orifice to flow through the latter. If di^d) is 
the indefinitely small change of depth, and dR^ the indefinitely 
small change in the radius of the cylindrical portion resting 
over the orifice, then the equality of volumes expressing this 
hypothesis is the following : 



Art. 1 02.] MERIDIAN SECTION. 72^ 

7t{B? - R/) . d{d) = 2nR^d . dR, ; 
or ; 

d(d) _ 2R, dR, 



d R' - R,' 



(2) 



3°. — T/ie rate of decrease of the radius of the primitive cen- 
tral cylinder is constant tJiroiighoiit its lengtJi at any given instant 
during flow. 

Let r be any radius less than R^y then if the latter is de- 
creased by the very small amount dR^^ the former will be 
shortened by the amount dr ; and by the last hypothesis there 
must result : 

dR. dr , . 

^ = y (3) 

This is a perfectly general equation, in which r may or may 
not be the variable value of the radius of that portion of the 
primitive central cylinder remaining above the orifice at any 
instant during flow. 

These are the three hypotheses on which Tresca based his 
theory of the flow of solids. It is thus seen to be put upon a 
purely geometrical basis, entirely independent of the elastic or 
other properties of the material. 



Art. 102. — The Variable Meridian Section of the Primitive Central 

Cylinder. 

The meridian curve haH, or hbK, Fig. 2 of Art. 100, may 
now easily be determined. 

Eq. (i) of Art. loi may take the first of the following forms, 
while its differential, considering d and y variable, may take 
the second ; 



y2S FLOW OF SOLIDS. [Art 102. 



d{d) ==-^dy. 



Dividing the second by the first : 

d{d) _ dy _ 2R^ dR^ 



d R^ ^ R^ - R^ 

y — FT— D 

•^ R^ 

The last member of this equation is simply Eq. (2) of Art. 
loi ; and if the value of dR^, in Eq. (3) of the same Article, 
be inserted in the third member of this equation, there will 
result : 

2R^ dr dy 



R^ - R^ ' r ^' r» * ' 

y D 

•^ R,' 

Integrating between the limits of r and R^, and remember- 
ing that r will be restricted to the representation of the radius 
of that portion of the primitive central cylinder which remains, 
at any instant, over the orifice, by takings = o for r = R^ : 



2R ^ r 




R" - R,' "^ R, 



^^ log'' indicates a Napierian logarithm. 

Passing from logarithms to the quantities themselves, and 
reducing : 



Art. 103.] 



HORIZON'TAL SECTIONS. 



729 



7 = 



R'D 



K' 



RJ 



2/?, 2 



/?.- 



(I) 



This is the desired equation of the line, in which r is meas- 
ured normal to the axis of the cylinder or jet, while/ is meas- 
ured along that axis from the extremity of the jet. When the 
material is wholly expelled : 



y 



R- 



D, and r = o. 



Eq. (2) is applicable to the jet only. For the line hF or Gk^ 
resort will had to the equation : 

d{d) _ 2R^ dr 
d ~ R" - R,^ ~7' 

Again integrating between the limits d and Z>, or r and R^y 
and reducing : 



2 



- ^- (S^ 



(2) 



This value of r is the radius of that portion of the primitive 
central cylinder which remains over the orifice when D is re- 
duced to d. 



Art. 103. — Positions in the Jet of Horizontal Sections of the Primitive 

Central Cylinder. 

That portion of the primitive central cylinder below ab in 
Fig. I of Art. 100, will be changed to abKH in Fig. 2 of the 
same Article. 

If, in the latter Fig., y' is the distance from HK to ab^ 



730 



FLOW OF SOLIDS. 



[Art. 103. 



measured along the axis, then the volume of HKab will have 
the value 

[y' 

Jo 

If d' is the distance aF—bGy in Fig. i, the equality of 
volumes will give : 



r^ dy = R,\D - d'). 



Eq. (i) of Art. 102 gives : 



r^ = R^ [\ - 



RW 



R^ 



... \^r^dy = R^^D - R,^d{i - ^^'^ = RA^ -d'). 



y 



R' 
R} 



I — 



d\li^ 
D 



D 



. . (I) 



If iVis the number of horizontal layers required to compose 
the total thickness D^ and n the number in the depth d' : 



Hence : 



^ R} 



d' 

D 


n 


— 


f ?i\r^ 


I — 


■\n) 



D 



(2) 



Art. 105.] PATH OF MOLECULE. 731 

Tresca computed values of y for some of his experiments, 
and compared the results with actual measurements. The 
agreement, though not exact, was very satisfactory. Within 
limits not extreme, the longer the jet the more satisfactory was 
the agreement. 



Art. 104.— Final Radius of a Horizontal Section of the Primitive Central 

Cylinder. 

Let it be required to determine what radius the section 
situated at the distance d' from the upper surface of the primi- 
tive central cylinder will possess in the jet. 

It will only be necessary to put for j in Eq. (i) of Art. 102, 
the value of y taken from Eq. (i) of Art. 103. This operation 
gives : 

d \ R^ fr \ R"^ - Ri^ 



Hence : 



nj ~ \R, 



r' 



R"^ - /?)'' 



If i?i is small, as compared with R, there will result ap- 
proximately : 



Art. 105. — Path of any Molecule. 

The hypotheses on which the theory of flow is based enable 

the hypothetical path of any molecule to be easily established. 

In consequence of the nature of the motion there will be 



732 FLOW OF SOLIDS. [Art. I05. 

three portions of the path, each of which will be represented 
by its characteristic equation, as follows : 

First : let the molecule lie outside of the primitive central 
cylinder. 

Let R and H be the original co-ordinates of the molecule 
considered, measured normal to and along the axis of the 
cylinder, respectively, from the centre of the orifice HKi^'x^. i 
Art. 100) as an origin, while r and h are the variable co- 
ordinates. 

The first hypothesis, by which the density remains con- 
stant, then gives the following equation : 



or : 



n<^R' - R'')H = n{E} - r^yi ; 
hR' - hr' = {R' - R'')II (i) 



This is the equation to the path of the molecule, in which 
r must always exceed R^, 

As this equation is of the third degree, the curve cannot be 
one of the conic sections. 

Second : let the molecule move in the space originally occupied 
by the central cylinder. 

While h and r now vary, the volume 7rr^[D — li) must re- 
main constant. When r = Rj^ let h =- h^. Hence : 

r\D - h) = R,\D - h,) (2) 

But if h = //j and r = R^'m Eq. (i) : 

Placing this value in Eq. (2) : 

^(D - A) = R,'(d -1/ p^) . ... (3) 



Art. 105.] PATH OF MOLECULE. 733 

Third : let the molecule move in the jet. 

After the molecule passes the orifice, its path will evidently 
be a straight line parallel to the axis of the jet. Its distance 
r^ from that axis will be found by putting h = o in Eq. (3). 
Hence : 

/ H R^ - R'^y^ 



ADDENDA. 



Addendum to Art. 20, 

* 
Some problems similar to that treated on pages 134 and 
135, but of a less simple character, arise in connection with the 
design of railway track stringers. The general method of solu- 
tion of such problems has already been indicated in Art. 20, 
but will be here applied to four equal weights, each being 
represented by W, 

Case L 

The relative position of these four weights is shown in Fig. 
I, in which / is the span and a^ c and b the distances separating 
the adjacent pairs of weights. The latter distances are fixed 
or constant. 






(^ '@ !(^(^ 





h 



-^ — N- ^T 

I ' \ 

Fig.l 



Since c and b are each greater than a, the shear will be 
zero under the weight A when all the weights are so placed as 



Art. 20.] ADDENDA. 735 

to give the beam its greatest possible bending moment. It 
will only be necessary, then, to find such a position of the 
loading as will make the bending moment under A the greatest 
possible. 

Let X be measured to the left from the right abutment, as 
shown ; then the left reaction at R will be : 

R = W {^^±^^±^^±±^ .... (I) 
Hence the bendinp; moment under A will be : 



M 



^ ^^ 4^ + 3^+2.+ ^ ^ (/-;.- ^ -C) - Wb. (2) 



Taking the first derivative in reference to x : 

•■• " = T-|"--4:^-i*- • • • (3) 

Inserting this value of x in Eq. (2) and indicating the re- 
sulting maximum moment by M^^ : 



M^ is the greatest bending moment to which the beam or 
stringer can ever be subjected, and it will be found at the dis- 
tance {/ — X — a — c) = {%l — yia — %c -\- yib) from either 
abutment. 



73^ ADDENDA. [Art. 20. 

But in Fig. i, if x^ is the distance from the right abutment 
to the centre of gravity of the entire load : 

x^ = Yil- Yza- y^c-^ Yzb (5) 

Hence, the centre of span is midway between the centre of 
gravity of the load and wheel A or point of greatest bending. 



Case 2, 

In this case let a = c. Making this change in Eq. (4), the 
value of the greatest moment becomes : 

M,= '^{l-^a + Uy-Wb ... (6) 



The distance of greatest bending from either abutment 
takes the value : 

x^ = y2l - y?>a ^ %b (7) 



Case 3. 
Let a z= c — b, M^ and x^ then take the values : 

^x-^(/-fy- Wa (8) 

x^ = yi - ya (9) 

In the case of an actual stringer, the equal weights Ware 
the weights on the driving wheels of a locomotive. 



Art. 75.] ADDENDA. 73/ 



Addendum to Art. 73. 

A butt joint with a set of single or double cover plates or 
butt straps may be formed in such a manner that the rivets 
and cover plates will take very nearly or exactly their proper 
proportional loads. Each set of cover plates is composed of 
a series uniformly decreasing in length, the longest of the 
series lying adjacent to the main plates or mem/bers joined. 
One row of rivets parallel to the joint is then put through each 
end of each cover plate, and, of course, also through those 
lying underneath. In this manner the number of rivets from 
the end of the longest or lowest cover plate to any section 
parallel to the joint is proportional to the sectional area of the 
covers against which they pull ; the joint is consequently of 
nearly uniform resistance. 

The number of butt straps or cover plates in a set depends 
upon the size of the members joined. 

In most cases the rivets cannot take exactly their propor- 
tional loads, for the reason that those portions of the members 
joined which lie within the limits of the joint are not of uni- 
form resistance, as the system of covers is. 



Addendum to Art. 75. 

Some recent experiments (March, 1883), on steel eye-bars 
manufactured by the Edge Moor Iron Co. under their patents, 
and tested to failure, show such interesting and successful re- 
sults that they should be noticed in a work of this character. 
The heads were of the general form shown on page 647, the 
front portion ABD having been described with the radius r = 
yiD ; they were not thickened. 

The following are the data and results of tests : 

47 



738 



ADDENDA. 



[Art. 75. 



/ = length between centres of pin holes ; 

d =■ diameter of pin ; 

D = ADj Fig. 3, of page 647 ; or twice CB ; 

n = ratio of pin diameter over width of bar ; 

e = percentage of excess of eye section AD over 
area of bar section ; 
£. L. = elastic limit in pounds per sq. in. ; 
U/t. = ultimate resistance in pounds per sq. in. ; 

^ = percentage of reduction of fracture section ; 

/ = percentage of stretch of original (/ — d). 

All bars were 3 x yf = 2.4375 sq. ins. in sectional area, 
and were of mild steel. 



NO. 


/. 


d. 


D. 


«. 


e. 


E.L. 


Lit. 


1- 


/• 




Ft. 


Ins. 


Ins. 


Ins. 














I 


5 







81 


1. 31 


57.2 


48,000 


71,400 


48.0 


13.8 


2 


5 


3 

4 


4i"d- 


8^- 


1.50 


41.5 


49,000 


69, 600 


48.6 


13.7 


3 


5 


ii 

4 


4i^^ 


8^^ 


1.50 


40.0 


45,400 


68,800 


43.7 


14.8 


4 


5 


5. 
4 


3 d- 


1% 


1.06 


51.6 


44,100 


64, 700 


44.0 


17.0 


5 


5 


-5._ 


3h 


1% 


1. 14 


38.0 


44,100 


70,900 


540 


8.0 


6 


5 


-,^<T 


3U- 


7I 


1.23 


33 2 


44,100 


73,000 


50.6 


7.5 


7 


4 


I|^ 


4V§ 


9;^ 


1.64 


55-5 


49,400 


70,100 


48.0 


15.8 


8 


4 


iH 


5A' 


9-^ 


1.72 


46.4 


44,100 


64, 700 


40.8 


16.5 


9 


4 


in 


5i tj 


9I 


1.89 


29.2 


42,000 


64,400 


42.0 


17-5 


10 


4 


iH 


3i i> 5 16 


l\ 9I 


1.2 1.8 


33.0 40.0 


44,100 


64,400 


48.0 


16.3 



All values in column e, except those for No. 10, are means 
of two (one for each end of the bar), but in no case did either 
of the latter vary more than one and one half per cent, from 
the mean. 

The pin holes were elongated from one quarter to one inch. 
. All bars broke in the body of the bar, and none nearer the 
centre of the eye than about ten inches. Half of them broke 
in the vicinity of the centre of the bar. 

With the exception of No. 4 it will be observed that e is 



Art. 75.] ADDENDA. 739 

much less than required by ordinary practice with iron bars. 
But it will also be observed that n is much larger than is usually 
found in bridge practice. With smaller values of n, larger ex- 
cesses {e) might be found necessary, as No. 4 would seem to 
indicate. In any event, however, the tests show that the 
homogeneous character of steel insures it against much of the 
injury which iron suffers in the upsetting orocess preceding 
the formation of the head. 



INDEX. 



A. 

PAGE 

* Actual energy of elasticity 95, g6 

Adhesion between bricks and mortar 364, 365 

Alloys of copper, tin and zinc in compression 387-389 

Alloys of copper, tin and zinc in tension 336, 339-341, 343, 346, 347 

Aluminium bronze in tension 343 

American Bridge Co. column 438, 439, 442, 448 

Angle iron column 445 

Angle irons as columns 475-477 

* Annealing, effect of, on wrought iron 245 

* Annealing, effect of, on steel 298, 313, 319, 331 

t Artificial stones in compression 393, 395, 396 

Artificial stones in tension 363, 364 



B. 

» Bauschinger's experiments on steel 333 

♦ Bauschinger's experiments on wrought iron • • • • 262-268 

Beam, continuous v 177 

Beam, one end fixed and other simply supported 182-184 

Beam, fixed at both ends 188-190 

Beam, non-continuous with uniform load 139, 174 

Beam, non-continuous with single weight 138, 174 

Beams, solid 514, 515 

Beams, solid, cast iron 518-520 

Beams, solid, cement, mortar and concrete 537-542 

Beams, solid, combined iron and steel 523? 524 

Beams, solid, copper, tin, zinc and alloys 524-526 

Beams, solid, practical formulae for 543> 544 

Beams, solid, steel 520-523 



742 INDEX. 



PAGE 



Beams, solid, stone 543 

Beams, solid, timber 526-536 

Beams, solid, wrought iron 515-518 

Bending by continuous normal load 679-680 

Bending by oblique forces 674-679 

Bending in riveted joints 609-612, 614, 615 

Bending moments 129, 132, 133 

Bending moments, greatest 137 

Bending moments, greatest or least 134 

Bending moments, greatest, with four weights 735, 736 

Bessemer steel, coefficient of elasticity 288, 290, 291 

Bessemer steel, ultimate resistance 301-303, 321 

Box beams 595, 596 

Bouscaren's experiments (columns) 446, 447, 453 

Brass, fine yellow, in compression .' 389 

Brass, in tension 336, 340-344 

Brass, red 350 

Brick in compression 397 

Brick in tension 364 

Buckling of latticed columns 458, 459 

Building stones, natural, in compression 398-402 

Built beams, steel 600, 601 

Built beams, wrought iron 578-601 

Bulging of plates 665-674 

Butt joints, double covers 606, 633-639 

Butt joints of uniform resistance 737 

Butt joints, single butt strap. See " Lap Joint." 



c. 

* Cables or ropes 648-654 

Cadmium 351.352 

Cantilever 132 

Cantilever with single load 172 

Cantilever with uniform load 142, 172 

Cast-iron columns, hollow round 470, 472 

Cast-iron columns, solid round 447, 470, 472 

Cast-iron flanged beams 554-559 

Cast iron in compression 37^ 

Coefficients of elasticity 377 

Ultimate resistance 37^, 379 

Effect of remelting 379 



INDEX. 743 



PAGE 



Cast iron in shearing 487-489, 493 

Cast iron in tension ; 276 

Coefficient of elasticity and elastic limit 276 

Ultimate resistance 279 

Effect of remelting 282 

Effect of continued fusion 284 

Repetition of stress 284 

Effect of high temperatures 286, 347 

Cast iron in torsion 487, 500-503 

Cast steel, ultimate tensile resistance 301, 302 

Cement in compression 390-392 

Cement mortar in compression 354, 390-392 

Cement mortar in tension 354, 359, 360 

Portland 360 

Various brands 354 

Cement, pure, in tension 353 

Variation of strength with age 355, 356, 360, 361 

Maclay's experiments 357, 359 

Grant's experiments 361 

Keene's cement 361 

Parian cement 361 

Portland cement 357, 359, 360 

Chain cables ... 652-654 

Channels as columns 455, 461, 462 

Chemical constitution of wrought iron 270 

Chrome steel, coefficient of elasticity 286 

Circular cylinder, torsion of 75, 76 

Coefficient of elasticity 3, 208, 209, 512 

Coefficient of elasticity, wrought iron in flexure 516, 517 

Coignet beton in compression 396 

Collapse of flues 65 5-659 

Columns, ends round, flat or fixed 191 

Columns, limit of application of flexure formulae 195, 196 

Columns, long, flexure of 190 

Columns, reduction of end sections 479 

Columns, with flat ends 193 

Columns, with one round and one flat end 194, 195 

Columns, with round ends 194 

Common column 449 

Common theory of flexure 122 

Common theory of flexure, unequal coefficients of elasticity 199 

Compression, greatest in bent beam 204 

Concrete in compression 393, 394 



744 INDEX. 

PAGE 

Connections 606 

Continuous beam .* 132 

Centra-flexure 131 

Co-ordinates, equations in polar 27 

Co-ordinates, equations in rectangular 14 

Co-or<^inates, equations in semi-polar 20 

Copper in tension 336, 337, 339, 340, 343-445, 349 

Copper, tin, zinc and alloys in flexure 525, 526 

Copper, tin, zinc, lead and alloys in compression 386-389 

Copper, tin, zinc, lead and alloys in shearing 487, 495 

Copper, tin, zinc, lead and alloys in torsion 487, 507-509 

Crucible steel, coefficient of elasticity 291 

Crucible steel, ultimate tensile resistance 294, 303, 324 

Crystallization of wrought iron 259 

Curved beams, flexure of 124-129, 143, 144 

Cylinders, thick, hollow 36, 38 

Cylinders, thin, hollow 36, 37 



D. 

Deflection by common theory of flexure 130 

Deflection of cast-iron flanged beams 558 

Deflection of wrought-iron T" beams 562, 563 

Diagonal riveted joints 642 

Diameter of rivets 618, 619, 625, 631, 632, 634, 635, 637, 639 

Distribution of stress in riveted joints 607-615 

Drilling holes in steel 325-332 

Driving and drawing spikes •'663, 664 

Ductility 211 



E. 

Efficiency of riveted joint 638, 639 

Elastic limit, elevation of 260 

Elastic limit, Phoenix iron specimens 240 

Elasticity .' I 

Elasticity, coefficient of 3 

Elasticity, energy of go 

Elasticity, limit of 4, 209 

Elliptical cylinder, torsion of . . . . 54 

English wrought iron 258 



INDEX, 745 



PAGE 



Eye-bar heads, formation of 645-647, 738 

Eye bars, steel, ultimate tensile resistance 297, 738 

Eye beams, wrought-iron 5^7-578 

Euler's formula 193 

Euler's formula adapted to columns 463-46S 

Euler's formula, limit of applicability 477-479 



F. 

Fatigue of cast iron 284, 285 

Fatigue of metals 708-722 

Fixed-end column 191, 193, 433, 437, 439, 441, 442 

Flanged beams of cast iron ; 554-559 

Flanged beams with equal flanges 564-601 

Flanged beams with unequal flanges 546-564 

Flat-end column. See " Fixed-end Column." 

Flexure..., 106 

Flexure by continuous normal load 676, 680 

Flexure by oblique forces 674-679 

Flexure, coefficients of elasticity for . 512 

Flexure, coefficients of elasticity unequal 199 

Flexure, common theory of 122 

Flexure, formulae for rupture 512 

Flexure, graphical method 196 

Flexure, normal stress in 112 

Flexure, theory of 106 

Flow of solids 723-733 

Flues, collapse of ^55-659 

Flusseisen 263, 266-268 

Fracture of wrought iron 258 



G. 

Glass in compression ' 389 

Glass in tension 352, 353 

Gold 351, 352 

Gordon's formula 409, 430-442 

Grant's conclusions 362 

Graphical method for flexure 196 

Gun bronze in compression 386 

Gun bronze in tension 342 



74^ INDEX. 



PAGE 



Gun metal in tension 341, 343, 346, 347 

Gun wire, steel 322 



H. 

• Hammering, effect on steel 247, 315-319 

•Hardening, effect on steel 314 

t Hardening, effect on wrought iron 247 

Hay steel, coefficient of elasticity 292, 293 

Hay steel, ultimate and elastic limits 299, 300 

Hemp ropes 648-652 

Hodgkinson's formula 193, 409, 469-472 

Hooke's law 2 



I. 

Influence of time on strains 719, 722 

Intensity of continuous load 136 



K. 

Keene's cement in tension 361 

Keystone column 438, 439, 441 

Kirkaldy's conclusions 272 



L. 

Laidley's tests of timber columns 483, 484 

Lanza's tests of timber columns 481, 482 

Lap joints 606, 616-632, 639 

Lateral strains 4 

Latticed columns 455-460 

Launhardt's formulae 715, 716 

Lead in compression 389 

Lead in tension 351, 352 

Length of test piece, influence of 230, 231 

Limit of elasticity 209 

Load, intensity of continuous 136 

Long column 371, 409, 430 



INDEX. 747 



PACK 



Long column, flexure of igo 

Longitudinal oscillations 100-105 



M. 

Martin steel shapes in tension 321 

Mill columns, timber 481, 482 

Modulus of resilience 97 

Modulus of rupture for flexure or bending 129, 515, 545 

Moment, greatest bending 137 

Moment, greatest or least bending 134 

Moments, bending 129, 132, 133 

Moment of inertia 410, 411 

Moment of inertia, angle section 417 

Moment of inertia, angle section, oblique axis 429 

Moment of inertia, box column 413, 414 

Moment of inertia, channel section, false 416 

Moment of inertia, channel section, true 426 

Moment of inertia, circular section 423 

Moment of inertia, column of plates and angles 415 

Moment of inertia, deck section 427 

Moment of inertia, eye section, false 421 

Moment of inertia, eye section, true 425 

Moment of inertia, latticed columns 418, 419 

Moment of inertia, Phoenix section 424 

Moment of inertia, rectangular sections 422, 423 

Moment of inertia, star section 421 

Moment of inertia, tee section 420 

Mortise holes, shearing behind 664, 665 

Muntz metal in tension 343, 347 



N. 

Neutral axis 114 

Neutral curve, cantilever 171 

Neutral curve, continuous beam 177 

Neutral curve, non-continuous beam 174 

Ne'itral curve, special cases 171 

Neutral surface .... 114 

Non-continuous beam 132 



74^ INDEX. 

O. 

PAGE 

Open-hearth steel in tension 297, 306, 311, 326, 327 

Oscillations, longitudinal 100-105 

Oscillations, torsional 78 

Overlap in riveted joint 615, 626, 627, 635 



P. 

Palladium 351, 352 

Parian cement in tension 354, 361 

Permanent set 211 

Phoenix columns 438, 439, 442, 448-455 

Phosphor bronze 344, 347 

Pin connection 644-648 

Pin end columns 437, 439 

Pitch of rivets 616, 617, 621, 623, 626, 629, 632-634, 637, 639 

Plates, bulging of 665, 674 

Platinum 351, 352 

Portland cement and mortar in tension 354, 357, 359, 360 

Potential energy of elasticity 94, 96 

Practical formulae for solid beams 543, 544 

Pressure on rivets 613, 619-624, 627-629, 632, 634-637 

Punching, effect of in riveted joints 615, 624, 634 

Punching steel, effect of 325-332 



Rail steel 295 

Reactions under continuous beams 157, 171 

Reaming holes in steel 325-332 

Rectangular cylinder, torsion of 59 

Red brass in tension 350 

Reduction of column ends 479 

Reduction of resistance between ultimate and breaking points 269 

Resilience 96, 97 

Resilience, modulus of 97 

Rivet steel 315 

Riveted joints 606-643 

Riveted joints, friction of 642 



INDEX. 749 



PAGE 



Riveted truss joints 640-642, 737 

Rollers, resistance of 659-662, 693, 694 

Rolling, eflfect of, on steel 315-319 

Ropes, iron, steel and hemp 648-652 

Round end columns 191, 194, 433, 439, 442 



s. 

Safety factor 681 

Sandberg's conclusions regarding low temperatures 253 

Shape iron in tension : 257 

Shape steel in tension 321 

Shapes, tensile resistance of ' 238 

Shear, counter 137 

Shear, greatest bending intensity in rectangular beam 121 

Shear, greatest total in non-continuous beam 137 

Shear, in bent beam 133 

Shear, main 137 

Shearing behind mortise holes 664, 665 

Shearing, coefficient of elasticity 4, 6, 487-490 

Shearing, greatest in bent beam 206 

Shearing of rivets 621, 624, 625, 630-632, 637 

Shearing, ultimate resistance 490-498 

Short blocks 371 

Siemens-Martin steel 324 

Siemen's steel 312 

Size of test piece, effect of 212, 224, 225 

Skin of bar, resistance of 233 

Solid beams, rectangular and circular 514, 544 

Spangenberg's experiments 709 

Specifications, Franklin Square bridge 694-698 

Specifications, Greenbush bridge 686-688 

Specifications, Menomonee bridge 689-694 

Specifications, Niagara suspension bridge. 688 

Specifications, Plattsmouth bridge 701-703 

Specifications, railway bridges 699, 700 

Specifications, Sabula bridge 682-6S6 

Specifications, steel cable wire, East River bridge 703-705 

Specifications, steel wire rope. East River bridge 705, 706 

Specifications, steel work for East River bridge 706, 707 

Spheres, thick, hollow 84 

Spikes, driving and drawing 663, 664 



750 INDEX. 



PAGE 



Square columns 438, 439, 441, 448 

Steel columns 448 

Steel in compression 380 

Coefficient of elasticity 380, 382, 383 

Elastic limit and ultimate resistance 380-385 

Effect of tempering 381 

Effect of annealing 382 

(For various grades and varieties see text under preceding heads.) 

Steel in shearing 487, 493-495 

Steel in tension 286 

Coefficient of elasticity 286 

Ultimate resistance and elastic limit 294 

Boiler plate 306 

Hardening and tempering steel plate 314 

Rivet steel 315 

Reduction of section by hammering and rolling 315 

* Annealing steel 319 

Steel wire , 319 

Shape steel 321 

Gun wire 322 

» Effect of low and high temperature on steel 323 

Constructive manipulations, such as punching, diilling and reaming. 325 

Bauschinger's experiments 333 

Fracture of steel 334 

Effect of chemical composition 334 

Steel in torsion 487, 504-507 

Steel plate, coefficient of elasticity 290, 291, 292 

Steel plate, ultimate tensile resistance 294, 306, 314, 326, 327 

Sterro-metal in tension 343 

Stones, natural building in compression 398-402 

Strain i 

Strains, influence of time on 719-722 

Strains, lateral 4 

Stress , I 

Stresses, expressions for tangential and direct 8 

Stresses, greatest at any point in a beam 202 

Stresses, plane of greatest normal in a beam 204 

Stresses, plane of greatest shearing in a beam 206 

Styffe's conclusions regarding low temperature 252 

Suddenly applied forces or loads 98, 100 

Suddenly applied stress, resistance of iron to 269 

Swelled columns 434, 439, 441 



INDEX. 75 1 

T. 

PACE 

Temperature, effect of increase on wrought iron 247-251, 324, 347 

Temperature, effect of low on wrought iron 251-254 

Temperatures, effect of high and low on steel 323, 324 

Tempering, effect on steel 31^ 

Tension, greatest in bent beam 204 

Theorem of three moments 146 

Theoreih of three moments, common form 154, 158 

Theory of flexure, general formulae 143 

Thick, hollow cylinders 36, 3S 

Thin, hollow cylinders 36, 37 

Thick, hollow spheres 84 

Thickness of web plate in flanged beams 601-603 

Thurston's conclusions regarding low temperatures 253 

Thurston's relation between tension and torsion 510 

Timber beams 526-536 

Timber beams of natural and prepared woods 536 

Timber columns, C. Shaler Smith's formulcC 485, 486 

Timber columns or pillars 471, 480-486 

Timber in compression 403-408 

Timber in shearing 487, 488, 496-498 

Timber in tension 3^5-370 

Timber in torsion 487, 4S8, 509, 510 

Tin in compression 3S7 

Tin in tension 336, 337, 339, 343 

Tobin's alloy in tension 336, 338, 339, 341 

Tobin's alloy in flexure 525, 526 

Torsion, coefficient of elasticity 8, 487-490, 498 

Torsion, general observations 77 

Torsion, greatest shear in circular sections 76 

Torsion, greatest shear in elliptical sections 56 

Torsion, greatest shear in rectangular sections 7i> 74, 75 

Torsion, greatest shear in triangular sections 59 

Torsion in equilibrium 43 

Torsion, moment of circular sections 76 

Torsion, moment of elliptical sections 55, 56 

Torsion, moment of rectangular sections 70, 74, 75 

Torsion, moment of triangular sections 58, 59 

Torsion of circular section ' 75 

Torsion of elliptical section 54 

Torsion of rectangular section 59 

Torsion of triangular section 57 



752 ■ INDEX. 



PAGE 



Torsion pendulum 82, 83 

Torsional oscillations 78 

Townsend's experiments, flow of solids 725 

Tresca's experiments, flow of solids 724 

Tresca's hypotheses, flow of solids 725-727 

Triangular cylinders, torsion of 57 

Tubes as columns 473, 475 

Tubes, collapse of 655-659 



u. 

Ultimate resistance 210 

Ultimate resistance, elevation of 260 

Ultimate resistance of wrought iron along or across fibres 242, 243 

Ultimate resistance, Phoenix iron specimens 240 

Unequal coefficients of elasticity, flexure with igg 



w. 

Web plate of flanged beam, thickness of 601, 603 

Welded joints '..... 643, 644 

Weyrauch's formulae 717, 718 

Whitworth's compressed steel 305 

Wire, brass 341, 344 

Wire, copper 341, 344 

Wire, Fairhairn's tests 257 

Wire, phosphor bronze 344 

Wire, Roebling's tests on wrought iron 254 

Wire ropes 648-652 

Wire, steel 319, 320 

Wire, steel gun 322 

Wire, Thurston's tests on old wrought iron 255 

Woehler's experiments 710-714 

Woehler's law 708 

Working stress 68 1 

Wrought-iron chain cables 652-654 

Wrought-iron columns, solid rectangular 447, 474, 475 

Wrought-iron columns, solid round '. 447, 470 

Wrought iron in compression 372 

Coefficient of elasticity 373, 374 

Elastic limit and ultimate resistance 374-37^ 



INDEX. 753 



PAGE 



Wrought iron in shearing 487, 491-493 

Wrought iron in tension 212 

Coefficient of elasticity. . . 212-223 

EfTect of size 212 

Rounds and flats 213 

Plates by Franklin Institute committee 215 

St. Louis bridge specimens 216 

Values of coefficient to ultimate resistance. 217 

Graphical representation 220 

Ultimate resistance and elastic limit 223 

Influence of size and dimensions 224-226 

Values for large bars 227 

Reduction of piles 228, 229 

Influence of length 230, 231 

Influence of skin of bar 233 

Shapes .... 233 

Large bars and rounds 234, 235 

Specimens from bars, plates and angles 237-239 

Boiler plate 24 1 

Effect of annealing 245 

Effect of hardening 247 

Variation of resistance with increase of temperature 247 

Effect of low temperature 251 

Iron wire 254 

Resistance of shape iron , . 257 

English wrought iron 258 

Fracture of wrought iron 258 

Crystallization of wrought iron 259 

Elevation of ultimate resistance and elastic limit 260 

Bauschinger's experiments on the change of elastic limit and coeffi- 
cient of elasticity ... 262 

Resistance of bar iron to suddenly applied stress 269 

Reduction of resistance between the ultimate and breaking point. . . 269 

Effect of chemical constitution 270 

Kirkaldy's conclusions 272 

Wrought iron in torsion 487, 49S-500 

Wrought iron in tee beams 559-5^3 



z. 

Zinc in compression 388 

Zinc in tension 336, 339, 343 

48 



J 6a8 



